and $L^{(l-1)}:=L$, giving an increasing filtration of $L$. We denote the
successive quotients by $L_{i}:=L^{(i)}/L^{(i-1)}$ and
$U_{i}:=U^{(i)}/U^{(i-1)}$ and $P_{i}:=P^{(i)}/P^{(i-1)}$.

  6. (6)

By restricting these filtrations to the semisimple part $R<L$, we obtain
groups denoted $R^{(i)}<L^{(i)}$ with successive quotients $R_{i}$. Similarly,
restricting the filtration to $H$ yields

$H^{(i)}=U^{(i)}\rtimes R^{(i)}<P^{(i)}$

with successive quotients $H_{i}=U_{i}\rtimes R_{i}$.

  7. (7)

Let $T^{(i)}:=\prod_{j\leq i}\lambda^{[j]}(\mathbb{G}_{m})=\prod_{j\leq
i}\lambda^{(j)}(\mathbb{G}_{m})\cong\mathbb{G}_{m}^{i}$ be the $i$ dimensional
subtorus of $T=Z(L)$ generated by the first $i$ 1-PS’s. We write
$\widehat{H}^{(i)}:=H^{(i)}\rtimes T^{(i)}$.

  8. (8)

The 1-PS $\lambda^{[i]}:\mathbb{G}_{m}\rightarrow P$ descends to a 1-PS
$\lambda_{i}:\mathbb{G}_{m}\rightarrow P/P^{(i-1)}$ of length two and the
unipotent radical of the associated parabolic $P(\lambda_{i})$ is isomorphic
to $U_{i}$. Furthermore, $\lambda_{i}$ grades $U_{i}$. We write
$\widehat{H}_{i}:=\widehat{H}^{(i)}/\widehat{H}^{(i-1)}\cong
H_{i}\rtimes\lambda_{i}(\mathbb{G}_{m})$.

The basic idea of quotienting-in-stages is to quotient inductively by the
groups $\widehat{H}_{i}$, since the stabilisers in these groups have nicer
properties than those of the full $\widehat{H}$ at once. Moreover, at each
stage of our quotienting process, we will use a different 1-PS in the central
torus $T=Z(L)$ to grade $U^{[i]}$; consequently, we will only need to consider
stabilisers for the semisimple part $R$ of $L$.

######  Remark 4.7.

These filtrations have length $l-1$, but one could filter $L$ and $R$ (and
thus also $P$ and $H$) in $l$ steps, using all $l$ of the block rows. However,
since the above filtrations are only required to quotient by the non-reductive
unipotent radical $U$, which naturally has a length $l-1$ row-filtration, we
have made the filtrations of $L$ and $R$ the same length by combining the
bottom two rows in the final step. Thus $R_{i}=\mathrm{SL}_{m_{i}}$ for $1\leq
i\leq l-2$ and $R_{l-1}=\mathrm{SL}_{m_{l-1}}\times\mathrm{SL}_{m_{l}}$.

####  4.2.2. Loci in $X$

We continue to use the various subgroups of $P$ from Definition 4.6 above.

######  Definition 4.8.

We define the following subsets of $X$, by analogy with Definition 2.13.

  1. (1)

The _$[i]$ -minimal weight space_ is
$Z^{[i]}_{\min}=Z(X,\lambda^{[i]})_{\min}$ and the _$[i]$ -attracting open
set_ is $X^{[i]}_{\min}:=X(\lambda^{[i]})_{\min}$, which are related by the
_$[i]$ -retraction_ $p_{[i]}:X^{[i]}_{\min}\rightarrow Z^{[i]}_{\min}$ sending
$x\mapsto\lim_{t\rightarrow 0}\lambda^{[i]}(t)\cdot x$. The corresponding
(semi)stable sets are $Z^{[i],(s)s}_{\min}:=(Z^{[i]}_{\min})^{R^{(i)}-(s)s}$
and $X^{[i],(s)s}_{\min}:=p_{[i]}^{-1}(Z^{[i],(s)s}_{\min})$.

  2. (2)

The _$(i)$ -minimal weight space_ is
$Z^{(i)}_{\min}=Z(X,\lambda^{(i)})_{\min}$ and the _$(i)$ -attracting open
set_ is $X^{(i)}_{\min}:=X(\lambda^{(i)})_{\min}$. We define the _$(i)$
-retraction_ $p_{(i)}:X^{(i)}_{\min}\rightarrow Z^{(i)}_{\min}$ sending
$x\mapsto\lim_{t\rightarrow 0}\lambda^{(i)}(t)\cdot x$, and the corresponding
(semi)stable sets $Z^{(i),(s)s}_{\min}:=(Z^{(i)}_{\min})^{R^{(i)}-(s)s}$ and
$X^{(i),(s)s}_{\min}:=p_{(i)}^{-1}(Z^{(i),(s)s}_{\min})$ in an analogous way.

  3. (3)

As a special case of (2), for $\lambda=\lambda^{(l-1)}$, the _$\lambda$
-minimal weight space_ is $Z_{\min}=Z(X,\lambda)_{\min}$ and the _$\lambda$
-attracting open set_ is $X_{\min}:=X(\lambda)_{\min}$, which are related by
the _$\lambda$ -retraction_ $p:X_{\min}\rightarrow Z_{\min}$ sending
$x\mapsto\lim_{t\rightarrow 0}\lambda(t)\cdot x$. The corresponding
(semi)stable sets are $Z^{(s)s}_{\min}:=(Z_{\min})^{R-(s)s}$ and
$X^{(s)s}_{\min}:=p^{-1}(Z^{(s)s}_{\min})$.

######  Remark 4.9.

As a rough conceptual shorthand, it may help the reader to bear in mind the
case of objects in an abelian category where $\lambda$ corresponds to a length
$l$ filtration: then $Z^{[i]}_{\min}$ is those objects for which the
filtration is split into two pieces _at the $i$th point_, and $Z^{(i)}_{\min}$
is those objects for which the filtration is split _at each of the first $i$
steps_. The maps $p_{[i]}$ and $p_{(i)}$ then correspond to taking a kind of
‘partial associated graded’; that is, an associated graded with respect to
certain coarsenings of the original filtration defined by $\lambda$ (see
$\S$4.2.4 below).

######  Remark 4.10.

  1. (1)

Note that here $Z^{(s)s}_{\min}$ is defined to be the (semi)stable set for the
semisimple group $R<L$ rather than the group
$\overline{L}=L/\lambda(\mathbb{G}_{m})$ as in Definition 2.13. If $T$ acts
trivially on $Z_{\min}$ (which will later follow from Assumption (QiS); see
Remark 4.32), then
$Z^{ss}_{\min}:=Z^{R-ss}_{\min}=Z^{\overline{L}-ss}_{\min}$, so for the
_semistable_ s locus it makes no difference which group we use; however, for
$l>2$, we have $Z^{\overline{L}-s}_{\min}=\emptyset$, while
$Z^{s}_{\min}:=Z^{R-s}_{\min}$ may be non-empty.

  2. (2)

We can also extend the definition of $\lambda^{[i]}$ in Definition 4.6 to
$i=l$. Then $\lambda^{[l]}$ is trivial and
$p_{[l]}:X=X^{[l]}_{\min}\rightarrow X=Z^{[l]}_{\min}$ is the identity map.

  3. (3)

If we are interested in an unstable HKKN stratum $S_{\beta}\cong
G\times^{P_{\beta}}Y_{\beta}^{ss}$ associated to the reductive group
$G=\mathrm{SL}_{N}$ acting on a projective variety $Y$, where the parabolic
subgroup $P_{\beta}$ is determined by a 1-PS $\lambda_{\beta}$ of $G$, then we
can apply this procedure to the $P_{\beta}$-action on
$X:=\overline{Y_{\beta}}$. Since $\lambda=\lambda_{\beta}$, we have that
$p:X_{\min}\rightarrow Z_{\min}$ is $p_{\beta}:Y_{\beta}\rightarrow Z_{\beta}$
and $Z^{ss}_{\min}:=Z_{\beta}^{R-ss}=Z_{\beta}^{ss}:=Z_{\beta}^{L_{\beta}-ss}$
if $Z(L_{\beta})$ acts trivially on $Z_{\beta}$.

####  4.2.3. The Quotienting-in-Stages stable set

Now we have introduced all this notation, we can define a subset of $X$ with
an explicit Hilbert–Mumford type description. We will eventually prove that
this set admits a geometric $P$-quotient under certain assumptions.

######  Definition 4.11.

The _Quotienting-in-Stages stable locus_ for the linearised $P$-action on $X$
is

$X^{P-qs}:=\left\\{\begin{array}[]{ll}&p_{[i]}(x)\notin
U^{[i-1]}Z^{[i-1]}_{\min}\text{ for all }2\leq i\leq l\\\ x\in
X^{s}_{\min}:&\dim\operatorname{Stab}_{U^{[i]}}(p_{[i]}(x))=d^{[i]}_{\min}\text{
for all }1\leq i\leq l-1\\\
&\dim\operatorname{Stab}_{U^{[i-1,i]}}(p_{[i]}(x))=d^{[i-1,i]}_{\min}\text{
for all }2\leq i\leq l-1\end{array}\right\\},$

where we recall $U^{[i-1,i]}:=U^{[i-1]}\cap U^{[i]}$ and, using the notation
of Definition 2.2, we set

$d^{[i]}_{\min}:=d_{\min}(X^{[i],ss}_{\min},U^{[i]})\quad\text{ and }\quad
d^{[i-1,i]}_{\min}:=d_{\min}(X^{[i],ss}_{\min},U^{[i-1,i]}),$

which are the generic stabiliser dimensions of $U^{[i]}$ and $U^{[i-1,i]}$ on
$X^{[i],ss}_{\min}$ respectively.

######  Remark 4.12.

We note that $U^{[i]}Z^{[i]}_{\min}=U^{(i)}Z^{[i]}_{\min}$, as $U^{[i]}\subset
U^{(i)}$ and the complement lies in the Levi $L^{[i]}$ of $P(\lambda^{[i]})$
and, as $\lambda^{[i]}$ is central in $L^{[i]}$, this Levi preserves
$Z^{[i]}_{\min}$.

A natural question is whether, or under what conditions, the locus $X^{P-qs}$
is non-empty. In Remark 4.45 below, we will see that, under the so-called
_Quotienting-in-Stages Assumption_ (QiS) and the _Upstairs Unipotent
Stabiliser Assumption_ (UU) introduced in §4.4.3, the locus $X^{P-qs}$ is a
non-empty open subset of $X_{\min}$. In fact, under the _Upstairs Unipotent
Stabiliser Assumption_ , the stabiliser conditions in this definition hold
automatically, and so

$X^{P-qs}=\\{x\in X^{s}_{\min}\>:\>p_{[i]}(x)\notin
U^{[i-1]}Z^{[i-1]}_{\min}\text{ for all }2\leq i\leq l\\}.$

####  4.2.4. Example: unstable sheaves

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 in $\S$3.2.1, we
consider the action of $P_{\tau}=P(\lambda_{\tau})=U_{\tau}\rtimes
L_{\tau}<\mathrm{SL}_{P(n)}$ on the closure $X:=\overline{Y_{\tau}}$ of
$Y_{\tau}$ in $Q_{n}$. Recall that the canonical linearisation
$\mathcal{L}_{\tau}$ is borderline for the grading of $U_{\tau}$ given by the
1-PS $\lambda_{\tau}$. The Quotienting-in-Stages procedure will determine a
small perturbation of this linearisation.

The (rational) 1-PS $\lambda_{\tau}$ determines a length $l$ filtration of
$V=\mathbb{C}^{P(n)}$

(9) $0=V^{(0)}\subset V^{(1)}=\mathbb{C}^{P_{1}(n)}\subset\cdots\subset
V^{(i)}=\mathbb{C}^{P_{\leq i}(n)}\subset\cdots\subset
V^{(l)}=\mathbb{C}^{P(n)}$

where $P_{\leq i}:=\sum_{j\leq i}P_{j}$ and $P_{>i}:=\sum_{j>i}P_{j}$. For a
quotient sheaf $q_{\mathcal{E}}:\mathcal{O}_{B}(-n)^{\oplus
P(n)}\twoheadrightarrow\mathcal{E}$ in $Q_{n}$, the filtration (9) induces a
length $l$ filtration of $\mathcal{E}$

(10)
$0=\mathcal{E}^{(0)}\subset\mathcal{E}^{(1)}\subset\cdots\subset\mathcal{E}^{(i)}=q_{\mathcal{E}}(V^{(i)}\otimes\mathcal{O}_{B}(-n))\subset\cdots\subset\mathcal{E}^{(l)}=\mathcal{E}$

and we write $\mathcal{E}_{i}:=\mathcal{E}^{(i)}/\mathcal{E}^{(i-1)}$ for the
successive subquotients; this filtration is the HN filtration of $\mathcal{E}$
if $q_{\mathcal{E}}\in Y_{\tau}^{ss}$. The limit of $\lambda(t)\cdot
q_{\mathcal{E}}$ as $t\rightarrow 0$ corresponds to the associated graded
sheaf $\operatorname{gr}(\mathcal{E}^{(\bullet)})=\oplus_{i}\mathcal{E}_{i}$
of the filtration (10); see [24, Lemma 4.4.3] and [23, $\S$5.1]. Thus the
morphism $p_{\tau}:Y_{\tau}^{ss}\rightarrow Z_{\tau}^{ss}$ takes a sheaf to
its HN-graded sheaf. In particular, we have the following statements
concerning the minimal weight space for $\lambda_{\tau}$ (see also
Propositions 3.5 and 3.7).

######  Lemma 4.13.

For the $P_{\tau}=P(\lambda_{\tau})$-action on $X:=\overline{Y_{\tau}}$, the
following statements hold.

  1. (1)

The retraction $p:X_{\min}\rightarrow Z_{\min}$ coincides with the retraction
$p_{\tau}:Y_{\tau}\rightarrow Z_{\tau}$.

  2. (2)

The semisimple part of $R_{\tau}=\Pi_{i=1}^{l}\mathrm{SL}_{P_{i}(n)}$ of
$L_{\tau}$ has stable locus

$Z_{\tau}^{s}:=(Z_{\tau})^{R_{\tau}-s}\cong Q^{s}_{n}(P_{1})\times\cdots\times
Q^{s}_{n}(P_{l})$

where $Q^{s}_{n}(P_{i})$ is the open subscheme of
$Q_{n}(P_{i}):=\operatorname{Quot}(\mathcal{O}_{B}(-n)^{\oplus
P_{i}(n)},P_{i})$ of quotient sheaves $q_{i}:\mathcal{O}_{B}(-n)^{\oplus
P_{i}(n)}\twoheadrightarrow\mathcal{E}_{i}$ such that $\mathcal{E}_{i}$ is
stable and $H^{0}(q_{i}(n))$ is an isomorphism.

  3. (3)

We have $X^{ss}_{\min}=Y_{\tau}^{ss}$ and
$X_{\min}^{s}=Y_{\tau}^{s}:=p_{\tau}^{-1}(Z_{\tau}^{s})$ is the locus of
quotient sheaves $q_{\mathcal{E}}\in Y_{\tau}^{ss}$ such that each subquotient
$\mathcal{E}_{i}$ in the HN filtration of $\mathcal{E}$ is stable.

Next consider the 1-PS $\lambda_{\tau}^{[i]}(t)$ with only two weights; this
induces length 2 filtrations

(11) $V^{[\bullet,i]}:=(0\subset V^{(i)}\subset V)\quad\text{ and
}\quad\mathcal{E}^{[\bullet,i]}:=(0\subset\mathcal{E}^{(i)}\subset\mathcal{E}),$

which are coarsenings of (9) and (10). The limit of $\lambda^{[i]}(t)\cdot
q_{\mathcal{E}}$ as $t\rightarrow 0$ corresponds to the associated graded
sheaf $\mathcal{E}^{(i)}\oplus\mathcal{E}/\mathcal{E}^{(i)}$.

######  Lemma 4.14.

For $m\gg 0$ depending on $\tau$, we have for all $1\leq i\leq l-1$, an a
$L_{\tau}$-equivariant isomorphism

$Z_{\min}^{[i]}\cong X\times_{Q_{n}(P)}(Q_{n}(P_{\leq i})\times
Q_{n}(P_{>i})).$

Moreover a quotient sheaf $q_{\mathcal{E}}\in Y_{\tau}^{ss}$ lies in
$U_{\tau}^{[i]}Z_{\min}^{[i]}$ if and only if
$\mathcal{E}\cong\mathcal{E}^{(i)}\oplus\mathcal{E}/\mathcal{E}^{(i)}$, i.e.
the inclusion of the $i$th sheaf in the HN filtration
$\mathcal{E}^{(i)}\subset\mathcal{E}$ is split.

###### Proof.

As outlined above, the components of the
$\lambda^{[i]}_{\tau}(\mathbb{G}_{m})$-fixed locus are products of two Quot
schemes, corresponding to the two sheaves in the associated graded of the
filtration induced by this 1-PS, and indexed by pairs of Hilbert polynomials
$(P^{\prime},P^{\prime\prime})$ such that $P=P^{\prime}+P^{\prime\prime}$. One
such component is given by the pair $(P_{\leq i},P_{>i})$ and we claim this is
the minimal weight space. Since $Y_{\tau}^{ss}$ is open in $X$, the minimal
weight for $\lambda_{\tau}^{[i]}$ acting on $X$ is the same as the minimal
weight for the action on $Y_{\tau}^{ss}$. For any $q_{\mathcal{E}}\in
Y_{\tau}^{ss}$, a quick calculation shows the Hilbert–Mumford weight of
$\lambda^{[i]}$ with respect to the twisted linearisation $\mathcal{L}_{\tau}$
is equal to zero; this computation follows from [24, Lemma 4.4.4] and is
similar to [22, $\S$4.3]. By taking $m$ sufficiently large, we can ensure that
the only pair of polynomials $(P^{\prime},P^{\prime\prime})$ giving this
minimum weight is $(P_{\leq i},P_{>i})$.

The final statement follows, as any sheaf quotient sheaf in
$U_{\tau}^{[i]}Z_{\tau,\min}^{[i]}=P_{\tau}Z_{\tau,\min}^{[i]}$ is isomorphic
to a sheaf in $Z_{\min}^{[i]}$ and this is precisely the locus where
$\mathcal{E}^{(i)}\subset\mathcal{E}$ is split. ∎

Similarly to Lemmas 4.13 and 4.14 one can prove the following result.

######  Lemma 4.15.

The minimal weight space for the 1-PS $\lambda^{(i)}_{\tau}$ admits the
following description.

  1. (1)

For $m\gg 0$ depending on $\tau$, we have for all $1\leq i\leq l-1$, an a
$L_{\tau}$-equivariant isomorphism

$Z_{\min}^{(i)}\cong X\times_{Q_{n}(P)}(Q_{n}(P_{1})\times\dots\times
Q_{n}(P_{i})\times Q_{n}(P_{>i})).$

Moreover a quotient sheaf $q_{\mathcal{E}}\in Y_{\tau}^{ss}$ lies in
$U_{\tau}^{(i)}Z_{\min}^{(i)}$ if and only if
$\mathcal{E}\cong\mathcal{E}_{1}\oplus\dots\mathcal{E}_{i}\oplus\mathcal{E}/\mathcal{E}^{(i)}$.

  2. (2)

The (semi)stable locus for the group $R^{(i)}$ are given by

$Z^{(i),(s)s}_{\min}:=(Z_{\min}^{(i)})^{R_{\tau}^{(i)}-(s)s}\cong
Q^{(s)s}_{n}(P_{1})\times\cdots\times Q^{(s)s}_{n}(P_{i})\times Q_{n}(P_{\geq
i}).$

Let us now determine the Quotienting-in-Stages stable locus in the case of
sheaves. Initially we ignore the conditions on unipotent stabilisers and focus
on the first part of the definition.

######  Proposition 4.16.

For $q_{\mathcal{E}}\in X^{s}_{\min}=Y_{\tau}^{s}$ and $2\leq i\leq l$, we
have $p_{[i]}(q_{\mathcal{E}})\in U_{\tau}^{[i-1]}Z^{[i-1]}_{\min}$ if and
only if $\mathcal{E}^{(i-1)}\subset\mathcal{E}^{(i)}$ is split.

###### Proof.

For $i<l$, we have $p_{[i]}(q_{\mathcal{E}})=q_{\mathcal{E}_{\leq i}}\oplus
q_{\mathcal{E}_{>i}}$ with corresponding quotient sheaf
$\mathcal{E}^{(i)}\oplus\mathcal{E}/\mathcal{E}^{(i)}$. The 1-PS
$\lambda^{[i-1]}_{\tau}(t)$ induces the following filtration of
$\mathcal{E}^{(i)}\oplus\mathcal{E}/\mathcal{E}^{(i)}$

$0\subset\mathcal{E}^{(i-1)}\subset\mathcal{E}^{(i)}\oplus\mathcal{E}/\mathcal{E}^{(i)},$

which is split (i.e. $p_{[i]}(x)\in U_{\tau}^{[i-1]}Z^{[i-1]}_{\min}$) if and
only if $\mathcal{E}^{(i-1)}\subset\mathcal{E}^{(i)}$ is split. For $i=l$, we
have $p_{[l]}=\mathrm{Id}$ and so $p_{[l]}(x)\in
U_{\tau}^{[l-1]}Z^{[l-1]}_{\min}$ if and only if
$\mathcal{E}^{(l-1)}\subset\mathcal{E}^{(l)}=\mathcal{E}$ is split. ∎

The next step is to describe the unipotent stabilisers for $U_{\tau}^{[i]}$
and $U_{\tau}^{[i-1,i]}$.

######  Proposition 4.17.

For any $q_{\mathcal{E}}\in Y^{ss}_{\tau}$, the following statements hold.

  1. (1)

For $1\leq i\leq l-1$, we have
$\operatorname{Lie}\operatorname{Stab}_{U_{\tau}^{[i]}}(q_{\mathcal{E}})\cong\operatorname{Hom}(\mathcal{E}/\mathcal{E}^{(i)},\mathcal{E}^{(i)})\cong\operatorname{Lie}\operatorname{Stab}_{U_{\tau}^{[i]}}(p_{[i]}(q_{\mathcal{E}}))$.

  2. (2)

For $2\leq i\leq l-1$,
$\operatorname{Lie}\operatorname{Stab}_{U_{\tau}^{[i-1,i]}}(q_{\mathcal{E}})\cong\operatorname{Hom}(\mathcal{E}/\mathcal{E}^{(i)},\mathcal{E}^{(i-1)})\cong\operatorname{Lie}\operatorname{Stab}_{U_{\tau}^{[i-1,i]}}(p_{[i]}(q_{\mathcal{E}}))$.

###### Proof.

Let us prove the first statement as the second follows similarly. Similarly to
Proposition 3.8 (i), we have
$\operatorname{Lie}\operatorname{Stab}_{U_{\tau}^{[i]}}(q_{\mathcal{E}})\cong\operatorname{End}_{-1}(\mathcal{E}^{[\bullet,i]})$
for the filtration
$\mathcal{E}^{[\bullet,i]}:=(0\subset\mathcal{E}^{(i)}\subset\mathcal{E})$.
Since this is a length 2 filtration, this filtered endomorphism group is the
homomorphism group from the quotient sheaf to the subsheaf, which gives the
first isomorphism. The second isomorphism follows as
$p_{[i]}(q_{\mathcal{E}})$ has corresponding quotient sheaf
$\mathcal{F}:=\mathcal{E}^{(i)}\oplus\mathcal{E}/\mathcal{E}^{(i)}$ and the
induced length two filtration on $\mathcal{F}$ is
$0\subset\mathcal{F}^{(i)}=\mathcal{E}^{(i)}\subset\mathcal{F}$ with
$\mathcal{F}/\mathcal{F}^{(i)}\cong\mathcal{E}/\mathcal{E}^{(i)}$. ∎

We can characterise the quotienting-in-stages stable locus for sheaves by
introducing a notion of $\tau$-stability for sheaves of HN type $\tau$
generalising the length 2 notion in [25].

######  Definition 4.18.

We say a sheaf of $\mathcal{E}$ of HN type $\tau$ is _$\tau$ -stable_ if the
following conditions hold:

  1. (1)

Each successive quotient $\mathcal{E}_{i}$ in the HN filtration is stable,

  2. (2)

Each inclusion $\mathcal{E}^{(i-1)}\subset\mathcal{E}^{(i)}$ in the HN
filtration is non-split,

  3. (3)

For $1\leq i\leq l-1$ and $2\leq j\leq l-1$ we have

$\dim\operatorname{Hom}(\mathcal{E}/\mathcal{E}^{(i)},\mathcal{E}^{(i)})=d_{\min}^{[i]}\quad\text{and}\quad\dim\operatorname{Hom}(\mathcal{E}/\mathcal{E}^{(j)},\mathcal{E}^{(j-1)})=d^{[j-1,j]}_{\min}.$

######  Corollary 4.19.

For, $q_{\mathcal{E}}\in Y_{\tau}^{ss}$, we have $q_{\mathcal{E}}\in X^{P-qs}$
if and only if $\mathcal{E}$ is $\tau$-stable.

###  4.3. The Quotienting-in-Stages procedure

Let us return to the more general case described in Definition 4.4, in which
$P<\mathrm{SL}_{N}$ is a parabolic subgroup of length $l$ acting on a
projective scheme $X$ with respect to an ample linearisation $\mathcal{L}$.
Our approach is to construct a quotient of this action in $l-1$ stages, where
we use $\lambda^{[i]}<T$ to grade the unipotent radical $U^{[i]}$ at the $i$th
stage. This gives an inductive procedure, where the base case and inductive
step look like quotienting by a group whose unipotent radical is the unipotent
radical of a maximal parabolic group; in particular, this unipotent radical is
abelian and so we can apply the results in $\S$2.3.7.

####  4.3.1. The base case and inductive step: quotients by length 2
parabolics

The base case and inductive step in the Quotienting-in-Stages procedure both
involve quotienting by a unipotent radical of a maximal (i.e. length $2$)
parabolic group, which is studied in [25]. If $l=2$, we have $P=P(\lambda)$
for a 1-PS $\lambda$ with two distinct weights $\beta_{1}>\beta_{2}$ with
multiplicities $(m_{1},m_{2})$. The central torus $T\cong\mathbb{G}_{m}$ of
$L$ is generated by $\lambda$, which grades the (abelian) unipotent radical
$U$. Thus we can directly apply the results of non-reductive GIT: in length
$2$, the surjection $R\rightarrow\overline{L}$ has finite kernel and so
(semi)stability for these groups coincides.

In length $l=2$, suppose that semistability coincides with stability for
$\overline{L}$ (or equivalently $R$); thus $Z^{s}_{\min}=Z^{ss}_{\min}$.
Additionally suppose that $[\widehat{U};\overline{L}\text{-ss}]$ holds (note
that as $U$ is abelian, this requires $\dim\operatorname{Stab}_{U}(-)$ to be
constant on $X^{ss}_{\min}$). Then by Theorem 2.29, we obtain a projective
geometric $P$-quotient

$q:X^{P-s}=X^{s}_{\min}\setminus UZ^{s}_{\min}\rightarrow X/\\!/P.$

If $[\widehat{U};\overline{L}\text{-ss}]$ fails in length $l=2$, let
$d_{\min}$ be the minimal dimension of $\dim\operatorname{Stab}_{U}(-)$ on
$X^{ss}_{\min}$ and we perform unipotent blow-ups. If
$[C(Z_{\min},U,d_{\min})\neq\emptyset]$ holds, we will not blow-up all of the
minimal weight space and by Proposition 2.37 (and [25]), the stable locus of
Definition 2.36,

$X^{\widehat{H}-\hat{s}}=\\{x\in X^{s}_{\min}\setminus
UZ^{s}_{\min}:\dim\operatorname{Stab}_{U}(p(x))=d_{\min}\\},$

admits a geometric and quasi-projective $\widehat{H}_{\beta}$-quotient. This
quotient is projective if additionally

$\dim\operatorname{Stab}_{U}(x)=d_{\min}\text{ for all }x\in Z_{\min}^{ss}.$

####  4.3.2. Quotienting-in-Stages for $l>2$

Recall the groups introduced in Definition 4.6. We will construct a quotient
in stages so that after stage $i$, we will have quotiented by the group
$\widehat{H}^{(i)}=H^{(i)}\rtimes T^{(i)}$, and to go from stage $i-1$ to
stage $i$, we take a quotient by the group
$\widehat{H}_{i}=H_{i}\rtimes\lambda_{i}(\mathbb{G}_{m})$, where the graded
unipotent part of this action $U_{i}\rtimes\lambda_{i}(\mathbb{G}_{m})$ is
induced by an action of $U^{[i]}\rtimes\lambda^{[i]}(\mathbb{G}_{m})$ on $X$.

######  Construction 4.20.

(_Quotienting-in-Stages Construction_)

Base stage: Let $X_{1}:=X$ and suppose that the action of
$\widehat{H}_{1}=\widehat{H}^{(1)}$ satisfies all the assumptions of Theorem
2.29, so we can take a NRGIT quotient giving a rational map

$q_{1}:X_{1}\dashrightarrow X_{2}$

to a projective scheme $X_{2}$, which restricts to a geometric quotient on its
domain of definition.

Inductive step: Assume we have constructed successive quotients
$q_{j}:X_{j}\rightarrow X_{j+1}$ of the $\widehat{H}_{j}$-action on $X_{j}$
for all $j<i$, whose composition

$q_{(i-1)}:=q_{i}\circ\cdots\circ q_{1}:X\dashrightarrow X_{i}$

is a $\widehat{H}^{(i-1)}$-quotient of an open subset of $X$. On $X_{i}$,
there is an induced linearised action of $\widehat{H}_{i}$ coming from the
linearised $\widehat{H}^{(i)}$-action on $X_{1}$ and assuming we can apply
Theorem 2.29, we can take a NRGIT quotient of this $\widehat{H}_{i}$-action

$q_{i}:X_{i}\dashrightarrow X_{i+1}$

and obtain a quotient of an open subset of $X$ by defining

$q_{(i)}=q_{i}\circ\cdots\circ q_{1}:X\dashrightarrow X_{i+1}$

to be the composition. Then $\operatorname{dom}(q_{(i)})\subset X$ is open
subset admitting a projective geometric $\widehat{H}^{(i)}$-quotient given by
the map $q_{(i)}$.

Final step (l-1): On the $\widehat{H}_{l-2}$-quotient $X_{l-1}$, there is an
induced linearised action of $\widehat{H}_{l-1}$ coming from the
$\widehat{H}^{(l-1)}$-action on $X$. Again assuming we can apply Theorem 2.29,
we can take a $\widehat{H}_{l-1}$-quotient $q_{l-1}:X_{l-1}\dashrightarrow
X_{l}$, and let

$q:=q_{(l-1)}=q_{l-1}\circ\cdots\circ q_{1}:X_{1}\dashrightarrow X_{l}$

be the composition. Thus
$q|_{\operatorname{dom}(q)}:\operatorname{dom}(q)\rightarrow X_{l}$ is a
projective geometric $\widehat{H}^{(l-1)}$-quotient.

If at each stage we are in the best case covered by Theorem 2.29, then the
domain of definition of each $q_{i}$ admits an explicit Hilbert–Mumford
description as the $\widehat{H}_{i}$-stable locus in $X_{i}$ given by
Definition 2.24. This procedure consists of $l-1$ quotients:

---
$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}\cdots\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{l-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{l}}$$\textstyle{X_{1}^{\widehat{H}_{1}-s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{1}}$$\textstyle{X_{2}^{\widehat{H}_{2}-s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2}}$$\textstyle{\cdots\cdots\cdots}$$\textstyle{X_{l-1}^{\widehat{H}_{l-1}-s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{l-1}}$

such that the composition $q:X\dashrightarrow X_{l}$ gives a projective
geometric $P$-quotient of its domain of definition.

In $\S$5, we will explicitly determine the domain of definition of $q$ under
certain assumptions. A large part of $\S$5 is devoted to understanding what
happens when the conditions of Theorem 2.29 fail at some stage. The basic idea
is to perform a sequence of (reductive or non-reductive) blow-ups on $X$
similarly to $\S$2.1.1 and $\S$2.3.6. This process results in a blow-up of $X$
for which the above construction can be carried out as written, and it remains
to relate the resultant stable locus on this blow-up with the original $X$.

Before we describe this sequence of quotients more carefully, we pause to
remark on the choices we are making at each stage.

######  Remark 4.21.

At each stage, one has to twist the induced linearisation by a (rational)
character in order for it to be well-adapted. Since the reductive groups
$R_{i}$ are products of special linear groups and have no non-trivial
characters, the twisting character can only be non-trivial on the grading 1-PS
$\lambda_{i}(\mathbb{G}_{m})$ and thus the space of twisting characters is
1-dimensional. In this 1-dimensional space of characters, only a small
interval will give characters that make the linearisation well-adapted. Thus,
there is in effect no choice of character111111Technically, at the next stage,
the possible choices of character needed to shift will depend on the previous
choices, but this is purely bookkeeping and does not effect the quotient.,
since we are always constrained to lie in a certain NRVGIT chamber in the
sense of [7]. In fact, we will soon see in §4.3.4 that, in the case we
consider, there is a single character of $\widehat{H}$ that induces well-
adapted characters of each $\widehat{H}_{i}$ at each stage. Hence, we may as
well choose this character from the beginning and fix the linearisation once
and for all, removing the need to twist at any subsequent point.

####  4.3.3. The Downstairs Stabiliser Assumption (D)

Our next task is to precisely state the assumptions we need at each stage in
order to apply Theorem 2.29 and execute Construction 4.20. First we need some
more notation.

######  Definition 4.22.

For the linearised $\widehat{H}_{i}$-action on $X_{i}$, we introduce the
following notation:

  1. (1)

The _stage $i$ minimal weight space_ is
$Z_{i,\min}:=Z(X_{i},\lambda_{i})_{\min}$,

  2. (2)

The _stage $i$ attracting open set_ is
$X_{i,\min}:=X_{i}(\lambda_{i})_{\min}$,

  3. (3)

The _stage $i$ retraction_ is $p_{i}:X_{i,\min}\rightarrow Z_{i,\min}$ is
given by $p_{i}(x):=\lim_{t\rightarrow 0}\lambda_{i}(t)\cdot x$,

  4. (4)

The _stage $i$ (semi)stable minimal weight space_ is
$Z_{i,\min}^{(s)s}:=Z_{i,\min}^{R_{i}-(s)s}$,

  5. (5)

The _stage $i$ (semi)stable locus_ is
$X_{i,\min}^{(s)s}:=p_{i}^{-1}(Z_{i,\min}^{(s)s})$.

With this notation introduced, we can state the conditions we will assume when
applying Construction 4.20.

######  Assumption 4.23.

(_The Downstairs Stabiliser Assumption_)

  1. (1)

We say the _downstairs unipotent stabiliser assumption_ (DU) holds if

(DU) $\forall\>1<i<l\>:\>\>\dim\operatorname{Stab}_{U_{i}}(-)\text{ is
constant on }X_{i,\min}^{ss},$

  2. (2)

We say the _downstairs reductive stabiliser assumption_ (DR) holds if

(DR) $\forall\>1<i<l\>:\>\>\dim\operatorname{Stab}_{R_{i}}(-)=0\text{ on
}Z_{i,\min}^{ss}.$

  3. (3)

We say the _$R$ -stable locus and semistable locus coincide_ if
$X^{ss}_{\min}=X^{s}_{\min}$, or equivalently:

$\dim\operatorname{Stab}_{R}(-)=0\text{ on }Z_{\min}^{ss}.$

We say the _Downstairs Stabiliser Assumption_ (D) holds if all the above
assumptions hold.

If (D) holds, Theorem 2.29 tells us that
$X_{i}^{\widehat{H}_{i}-s}=X_{i,\min}^{s}\setminus U_{i}Z_{i,\min}^{s}$ admits
a projective geometric $\widehat{H}_{i}$-quotient
$q_{i}:X_{i}^{\widehat{H}_{i}-s}\rightarrow X_{i+1}$.

######  Definition 4.24.

If (D) holds, we define the _Quotienting-in-Stages map_

$q:=q_{(l-1)}:=q_{l-1}\circ\cdots\circ q_{1}:X=X_{1}\dashrightarrow X_{l}$

to be the composition of the rational maps obtained in the above construction.
The following restriction of $q$ will also play an important role in our
argument: we further define

$q^{s}=q\mid_{X^{s}_{\min}}X^{s}_{\min}\dashrightarrow X_{l}$

to be the restriction of $q$ to $X^{s}_{\min}$. In the same way, we define
each $q^{s}_{(i)}$ to be the restriction of $q_{(i)}$ to $X^{s}_{\min}$, and
$q_{i}^{s}$ to be the restriction of $q_{i}$ to the open locus
$q_{(i-1)}(X^{s}_{\min})\subset X_{i}$.

One of our main aims for the remainder of this paper will be to show that,
under certain assumptions, the domain of definition of $q$ is equal to the
more explicit quotienting-in-stages stable locus $X^{P-qs}$ defined at
Definition 4.11.

######  Remark 4.25.

We will later show in Corollary 4.28 that $\operatorname{dom}(q)\subset
X^{s}_{\min}$, The latter will follow from the fact that $T$ acts trivially on
$Z_{\min}$, which is a consequence of Assumption (QiS) below.

The results of this subsection are summarised in the following corollary.

######  Corollary 4.26.

If (D) holds, then Construction 4.20 yields an open subset
$\operatorname{dom}(q)\subset X$ that has a projective geometric $P$-quotient.

###### Proof.

Since each $q_{i}$ is a geometric quotient of its domain of definition by
Theorem 2.29, the morphism $q:X\dashrightarrow X_{l}$ gives a projective
geometric $P$-quotient of $\operatorname{dom}(q)\subset X$. ∎

####  4.3.4. Constructing the quotient from a single character

Recall the discussion on twisting the linearisation at each stage in Remark
4.21. We will now prove that, under the hypothesis (D), we can find a single
character $P$ such that the Quotienting-in-Stages morphism $q$ is given by
taking invariant sections of this twisted linearisation.

The character groups of $P$ and its Levi $L$ are isomorphic; let us explicitly
write down the latter. Recall that the 1-PS $\lambda$ has $l$ distinct weights
$\beta_{1}>\dots>\beta_{m}$ with multiplicities $m_{i}$ and
$L\cong(\prod_{i=1}^{l}\mathrm{GL}_{m_{i}})\cap\mathrm{SL}_{N}$. Any
(rational) character $\chi:L\rightarrow\mathbb{G}_{m}$ is given by
$(r_{1},\dots,r_{l})\in\mathbb{Q}^{l}$ satisfying
$\sum_{i=1}^{l}m_{i}r_{i}=0$; namely, we have

$\chi(g_{1},\dots,g_{l})=\prod_{i=1}^{l}\det(g_{i})^{r_{i}}.$

Let us write $\chi_{\lambda}$ for the character corresponding to the rational
tuple $(\beta_{1},\dots,\beta_{l})$ of weights of $\lambda$. Let
$\mathcal{L}_{0}$ be obtained from $\mathcal{L}$ by twisting by a suitable
rational multiple of $\chi_{\lambda}$ to obtain a borderline linearisation in
the sense of Definition 2.16. For any character $\chi$, let
$\mathcal{L}_{\chi}$ be the linearisation obtained by twisting
$\mathcal{L}_{0}$ by $\chi$.

######  Proposition 4.27.

If (D) holds, then there is a small rational character
$\chi:P\rightarrow\mathbb{G}_{m}$ such that $\mathcal{L}_{\chi}$ induces well-
adapted linearisations at each stage of Construction 4.20. Hence, we have:

  1. (1)

The ring of $P$-invariant sections is finitely generated:

$R(X,\mathcal{L}_{\chi})^{P}\subset R(X,\mathcal{L}_{\chi}):=\bigoplus_{r\geq
0}H^{0}(X,\mathcal{L}_{\chi}^{\otimes r})$

  2. (2)

The Quotienting-in-Stages map $q:X=X_{1}\dashrightarrow X_{l}$ coincides with
the map

$q_{H,\chi}:X\dashrightarrow
X/\\!/_{\mathcal{L}_{\chi}}H:=\operatorname{Proj}(R(X,\mathcal{L}_{\chi})^{P}).$

In particular, this latter map is a good $P$-quotient of its domain of
definition

$X^{P-ss}(\mathcal{L}_{\chi}):=\\{x\in X:\exists\sigma\in
H^{0}(X,\mathcal{L}_{\chi}^{\otimes r})^{P}\text{ for }r>0\text{ with
}\sigma(x)\neq 0\\}.$

  3. (3)

Furthermore, $\operatorname{dom}(q)=X^{P-ss}(\mathcal{L}_{\chi})\subset
X^{P-ss}(\mathcal{L}_{0})=p^{-1}(Z^{\overline{L}-ss}_{\min})$ and we obtain a
map

$X_{l}\cong X/\\!/_{\mathcal{L}_{\chi}}H\rightarrow
X/\\!/_{\mathcal{L}_{0}}H\cong Z_{\min}/\\!/_{\mathcal{L}_{0}}L.$

###### Proof.

Since (D) holds, we do not need to perform any blow-ups in the Quotienting-in-
Stages construction and thus, at each stage, we just need to perturb the
linearisation by twisting by a character of the quotienting groups
$\widehat{H}_{i}$ to make it well-adapted, so that we can apply Theorem 2.29
at each stage. It thus suffices to show the choices of rational character used
to perturb the borderline linearisation $\mathcal{L}_{0}$ at each stage can be
lifted to a single choice of rational character
$\chi:P\rightarrow\mathbb{G}_{m}$ such that

$X_{l}\cong\operatorname{Proj}(R(X,\mathcal{L}_{\chi})^{P}).$

Once we have shown this, we already know $q:X\rightarrow X_{l}$ is a
surjective geometric (hence categorical) $P$-quotient, and the remaining parts
then follow from Proposition 2.18.

To prove the existence of the character $\chi$, we note that the quotienting
group $\widehat{H}_{i}$ at each stage in the Quotienting-in-Stages procedure
has 1-dimensional character group, arising form the 1-dimensional torus
$\lambda_{i}(\mathbb{G}_{m})$ which descends from
$\lambda^{[i]}(\mathbb{G}_{m})<P$. Let $\chi_{i}$ be the generator of the
character group of $\widehat{H}_{i}$ which is dual to the 1-PS $\lambda_{i}$.
Then at each stage we take a twist of the induced linearisation by small
rational multiple $\epsilon_{i}\chi_{i}$, with the choice of $\epsilon_{i}$
depending on the choices of $\epsilon_{j}$ required for well-adaptedness for
$j<i$. Recall from Definition 4.6 (1) that the 1-PS $\lambda$ for which
$P=P(\lambda)$ corresponded to weights $\beta_{1}>\dots>\beta_{l}$ with
multiplicities $m_{i}$ and the 1-PS $\lambda^{[i]}$ has two weights
$\beta_{\leq i}>\beta_{>i}$ with multiplicities $m_{\leq i}$ and $m_{<i}$. We
need to find a rational character $\chi_{\epsilon}$, corresponding to a vector
$(r_{1}(\epsilon),\dots,r_{l}(\epsilon))\in\mathbb{Q}^{l}$ with
$\sum_{i=1}^{l}m_{i}r_{i}(\epsilon)=0$, such that for $1\leq i\leq l-1$ we
have

$\epsilon_{i}=:<\chi_{i},\lambda_{i}>_{\mathfrak{t_{i}}}=<\chi_{\epsilon},\lambda^{[i]}>_{\mathfrak{t}}:=\beta_{\leq
i}\left(\sum_{j\leq
i}m_{j}r_{j}(\epsilon)\right)+\beta_{>i}\left(\sum_{j>i}m_{j}r_{j}(\epsilon)\right)$

where $\mathfrak{t}=\operatorname{Lie}(T)$ and
$\mathfrak{t_{i}}=\operatorname{Lie}(T_{i})$ for
$T_{i}:=\lambda_{i}(\mathbb{G}_{m})$. If we set $\epsilon_{0}=\epsilon_{l}=0$,
this system of equations admits the solution

$r_{i}(\epsilon):=\frac{1}{m_{i}}\left(\frac{\epsilon_{i}}{\beta_{\leq
i}-\beta_{>i}}-\frac{\epsilon_{i-1}}{\beta_{\leq
i-1}-\beta_{>i-1}}\right)\text{ for }1\leq i\leq l,$

where we note that the denominators appearing in this formula are non-zero. ∎

######  Corollary 4.28.

If (D) holds and $T$ acts trivially on $Z_{\min}$, then
$\operatorname{dom}(q)\subset X_{\min}^{s}$.

###### Proof.

Since $T=Z(L)\cong\mathbb{G}_{m}^{l-1}$ acts trivially on $Z_{\min}$, we have
$Z^{\overline{L}-ss}_{\min}=Z^{R-ss}_{\min}=:Z^{ss}_{\min}$ and this in turn
is equal to $Z^{s}_{\min}$ by the third part of (D). Then the result follows
from the last statement in Proposition 4.27 above as
$X_{\min}^{s}:=p^{-1}(Z^{s}_{\min})$. ∎

In the next section, we will see that the Quotienting-in-Stages Assumption
(QiS) implies that $Z(L)$ acts trivially on $Z_{\min}$, allowing us to apply
this Corollary.

###  4.4. The Quotienting-in-Stages and Upstairs Stabiliser Assumptions

In this section, we introduce the assumptions appearing in Theorem 1.2, which
we arrange into two groups, called the _Quotienting-in-Stages Assumption_ 4.29
(QiS) and the _Upstairs Stabiliser Assumption_ 4.40 (U) (or its weakened form
(WUU)).

Assumption (QiS) will be assumed to hold throughout, and is used to obtain an
explicit description of the domain of definition of the Quotienting-in-Stages
map. In contrast to this, the _Upstairs Stabiliser Assumption_ (U) specifies
ideal conditions on the stabilisers in $X$ under which we can use non-
reductive GIT to obtain a projective geometric quotient. If (U) fails, we can
perform a sequence of blow-ups and, under the so-called _Weak Upstairs
Unipotent Stabiliser Assumption_ (WUU), we obtain an explicit geometric and
quasi-projective quotient of $X^{P-qs}$ (see Theorem 5.25, which completes the
proof of Theorem 1.2).

####  4.4.1. The Quotienting-in-Stages Assumption

Here we use the notation from Definition 4.8.

######  Assumption 4.29.

(_The Quotienting-in-Stages Assumption_) We say that:

  1. (1)

_The minimal weight spaces are filtered_ if $Z^{(j)}_{\min}\subset
Z^{(i)}_{\min}\subset Z^{[i]}_{\min}$ for all $1\leq i<j\leq l-1$.

  2. (2)

_The preimages of $[i]$-minimal weight spaces are filtered_ if for all
$0<i<j<l-1$

$p_{[j+1]}^{-1}(Z^{[i]}_{\min}\cap Z^{[j+1]}_{\min})\subset
p_{[j]}^{-1}(Z^{[i]}_{\min}\cap Z^{[j]}_{\min});$

that is, for all $x\in X$, if $p_{[j+1]}(x)\in Z^{[i]}_{\min}\cap
Z_{\min}^{[j+1]}$, then $p_{[j]}(x)\in Z^{[i]}_{\min}\cap Z_{\min}^{[j]}$.

  3. (3)

_Non-degeneracy_ holds: there exists $x\in X^{s}_{\min}$ with
$p_{[j]}(x)\notin U^{[j-1]}Z^{[j-1]}_{\min}$ for all $2\leq j\leq l$.

When all these conditions hold, we say the _Quotienting-in-Stages Assumption_
(QiS) holds.

######  Remark 4.30.

The first two assumptions are modelled on the behaviour of unstable HKKN
strata for GIT problems associated to moduli of objects in an abelian
category. The final assumption, non-degeneracy, is equivalent to the non-
emptiness of the Quotienting-in-Stages stable set when (QiS) holds. If non-
degeneracy fails, we should interpreted this as the Hilbert-Mumford criterion
giving an empty semistable locus. For moduli of objects in an abelian
category, non-degeneracy is a property of the relevant Harder-Narasimhan type
which requires the existence of an object for which all extensions in the
Harder–Narasimhan filtration are non-split. In the sheaf case, we interpret
these assumptions in $\S$4.4.2: we prove that the first two assumptions hold
for unstable sheaves and give a sheaf-theoretic interpretation of non-
degeneracy.

######  Example 4.31.

It should be noted that (QiS) is not always satisfied, even for the problem of
performing quotients of unstable HKKN strata. Indeed, if we consider quintic
plane curves $C\subset\mathbb{P}^{2}$ up to the action of $\mathrm{SL}_{2}$,
with the Killing form and the natural linearisation, we see that there is an
HKKN stratum for which

$Z_{\beta}=\langle Y^{5},X^{4}Y\rangle,$

and the associated Kempf one-parameter subgroup $\lambda_{\beta}$ has (up to
scaling) weight vector $(2,1,-3)$ with respect to the natural coordinates.
Thus we have $\lambda^{[1]}$ with weight vector $(2,-1,-1)$ up to scaling, and
$\lambda^{[2]}$ has weight vector a scalar multiple of $(1,1,-2)$. Since the
length of $\lambda_{\beta}$ is $3$, we have $\lambda^{(2)}=\lambda_{\beta}$.
It is then easy to see that Assumption (1) of (QiS) is not satisfied, as

$Z^{[2]}_{\min}=\langle X^{4}Z\rangle\quad\text{and}\quad
Z^{[1]}_{\min}=\langle Y^{5}\rangle.$

Before we proceed, let us note some straight-forward consequences of (QiS).

######  Remark 4.32.

One immediate consequence of Assumption (1) in (QiS) is that $T$ acts
trivially on $Z_{\min}$. Indeed, this torus is generated by
$\lambda^{[i]}(\mathbb{G}_{m})$ for $1\leq i\leq l-1$, and we have
$Z_{\min}\subset Z^{[i]}_{\min}$ for all such $i$, where each $Z^{[i]}_{\min}$
is fixed pointwise by $\lambda^{[i]}(\mathbb{G}_{m})$.

######  Lemma 4.33.

Suppose as in Assumption (1) of (QiS) that $Z_{\min}\subset Z_{\min}^{[i]}$
for $1\leq i\leq l-1$. Then we have:

  1. _(i)_

$X_{\min}\subset X_{\min}^{[i]}$ and, for $x\in X_{\min}$, we have $(p\circ
p_{[i]})(x)=p(x)$,

  2. _(ii)_

For $x\in X_{\min}$, we have $(p_{[i]}\circ p_{[j]})(x)=(p_{[j]}\circ
p_{[i]})(x)$ for all $j\leq i$,

  3. _(iii)_

$Z_{\min}^{(s)s}\subset Z_{\min}^{R^{(i)}-(s)s}\subset Z_{\min}^{[i],(s)s}$,

  4. _(iv)_

$X_{\min}^{(s)s}\subset X_{\min}^{[i],(s)s}$,

  5. _(v)_

If $x\in X_{\min}^{(s)s}$, then $p_{[i]}(x)\in X_{\min}^{(s)s}$.

###### Proof.

Note that the images of the 1-PSs $\lambda^{[i]}$ and $\lambda$ lie in the
cental torus $T=Z(L)$. Using a sufficiently large power of our ample
linearisation on $X$ we can embed $X$ into $\mathbb{P}^{N}$ equivariantly and
choose coordinates on $\mathbb{P}^{N}$ so that the $T$-action is diagonalised.
Then we can evaluate the weights of any 1-PSs of $T$ by pairing the
$T$-weights with this 1-PS. By definition, $Z^{[i]}_{\min}$ is the minimal
weight space for $\lambda^{[i]}$ (in $X$) and $X^{[i]}_{\min}$ is the locus of
points in $X$ with at least one coordinate in $\mathbb{P}^{N}$ having this
minimal weight. Then $p_{[i]}:X^{[i]}_{\min}\rightarrow Z^{[i]}_{\min}$ is the
projection onto this minimal weight space. Assumption (1) means all the
$T$-weights that are minimal for $\lambda$ are minimal for $\lambda^{[i]}$ for
any $i$. Hence we can choose coordinates on $\mathbb{P}^{N}$ such that $p$ is
the projection on the first $n$ coordinates and $p_{[i]}$ is the projection
onto the first $m$ coordinates with $n\leq m$. In particular, this shows (i).
Similarly (ii) follows from Assumption (1).

For (iii), as $R^{(i)}<R$, we have $Z_{\min}^{(s)s}\subset
Z_{\min}^{R^{(i)}-(s)s}$ and then the claim follows from (i) together with the
functoriality of the semistable set with respect to equivariant closed
immersions. For (iv), if $x\in X^{(s)s}_{\min}$, then $p(x)\in
Z_{\min}^{(s)s}\subset Z_{\min}^{[i],(s)s}$ by (iii); therefore, by applying
Lemma 4.34 to $p$, we conclude that $p_{[i]}(x)\in Z_{\min}^{[i],(s)s}$ as
required.

Finally for (v), we have that $p(p_{[i]}(x))=p(x)\in Z^{(s)s}_{\min}$ using
(i), and so $p_{[i]}(x)\in X_{\min}^{(s)s}$. ∎

To complete the proof, we state the following result from about the behaviour
of semistable loci in reductive GIT under equivariant pullbacks.

######  Lemma 4.34.

[29, Chapter 1 §5] Let $G$ be a reductive group acting linearly on polarised
schemes $(X,\mathcal{L}_{X})$ and $(Y,\mathcal{L}_{Y})$ and let
$f:X\rightarrow Y$ is a $G$-equivariant morphism such that
$f^{*}\mathcal{L}_{Y}\cong\mathcal{L}_{X}$. Then $f^{-1}(Y^{G-ss})\subset
X^{G-ss}$, and if $f$ is quasi-affine we also have $f^{-1}(Y^{G-s})\subset
X^{G-s}$.

######  Remark 4.35.

By an almost identical proof to Lemma 4.33, one can show from the Assumption
(1) of (QiS) that $Z^{(i)}_{\min}\subset Z^{(j)}_{\min}\subset Z^{[j]}_{\min}$
for $1\leq j\leq i\leq l-1$ that the following statements hold.

  1. _(i)_

$X_{\min}^{(i)}\subset X_{\min}^{(j)}$ and for $x\in X_{\min}^{(i)}$, we have
$p_{(i)}\circ p_{(j)}(x)=p_{(i)}(x)$,

  2. _(ii)_

$X_{\min}^{(i)}\subset X_{\min}^{[j]}$ and for $x\in X_{\min}^{(i)}$, we have
$p_{(i)}\circ p_{[j]}(x)=p_{(i)}(x)$ and $p_{[i]}\circ p_{[j]}(x)=p_{[j]}\circ
p_{[i]}(x)$,

  3. _(iii)_

$Z_{\min}^{(i),(s)s}\subset Z_{\min}^{(j),(s)s}$ and
$Z_{\min}^{(i),(s)s}\subset Z_{\min}^{[j],(s)s}$,

  4. _(iv)_

$X_{\min}^{(i),(s)s}\subset X_{\min}^{(j),(s)s}$ and
$X_{\min}^{(i),(s)s}\subset X_{\min}^{[j],(s)s}$,

  5. _(v)_

If $x\in X_{\min}^{(i),(s)s}$, then $p_{(j)}(x)$ and $p_{[j]}(x)\in
X_{\min}^{(i),(s)s}$.

######  Remark 4.36.

One can generalise Lemma 4.33 to higher weight spaces and associated
attracting subschemes, which are higher Białynicki-Birula strata, which could
be useful for constructing a quotient when non-degeneracy fails.

####  4.4.2. An interpretation of (QiS) for sheaves of fixed HN type

Let us return to our running example of sheaves of fixed HN type
$\tau=(P_{1},\dots,P_{l})$ on $(B,\mathcal{O}_{B}(1))$. As in $\S$4.2.4, we
consider the action of $P_{\tau}$ on $X:=\overline{Y_{\tau}}\hookrightarrow
Q_{n}$ linearised by $\mathcal{L}_{\tau}$.

######  Proposition 4.37.

Parts (1) and (2) of (QiS) hold for the linearised $P_{\tau}$-action on
$X:=\overline{Y_{\tau}}$; that is the following statements hold.

  1. (1)

For all $0<i\leq j<l$, we have $Z^{(j)}_{\min}\subset Z^{(i)}_{\min}\subset
Z^{[i]}_{\min}$.

  2. (2)

For all $0<i<j<l-1$, we have

$p_{[j+1]}^{-1}(Z^{[i]}_{\min}\cap Z^{[j+1]}_{\min})\subset
p_{[j]}^{-1}(Z^{[i]}_{\min}\cap Z^{[j]}_{\min}).$

###### Proof.

The first claim follows immediately from the description of
$Z_{\min}=Z_{\tau}$ and $Z^{[i]}_{\min}$ and $Z^{(i)}_{\min}$ given
respectively in (5) and Lemmas 4.14 and 4.15 above.

For the second, let $0<i<j<l-1$. If $q_{\mathcal{E}}\in X^{[j+1]}_{\min}$,
then $p_{[j+1]}(q_{\mathcal{E}})=q_{\mathcal{E}_{\leq j}}\oplus
q_{\mathcal{E}_{>j+1}}$ has corresponding quotient sheaf
$\mathcal{F}:=\mathcal{E}^{(j+1)}\oplus\mathcal{E}/\mathcal{E}^{(j+1)}$ and
similarly $p_{[j]}(q_{\mathcal{E}})$ has corresponding quotient sheaf
$\mathcal{G}:=\mathcal{E}^{(j)}\oplus\mathcal{E}/\mathcal{E}^{(j)}$. As at
(11), the 1-PS $\lambda_{\tau}^{[i]}(t)$ induces filtrations

$\mathcal{F}^{[\bullet,i]}=(0\subset\mathcal{F}^{(i)}=\mathcal{E}^{(i)}\subset\mathcal{F})\quad\text{and}\quad\mathcal{G}^{[\bullet,i]}=(0\subset\mathcal{G}^{(i)}=\mathcal{E}^{(i)}\subset\mathcal{G}).$

If $p_{[j+1]}(q_{\mathcal{E}})\in Z^{[i]}_{\min}$, then the inclusion
$\mathcal{E}^{(i)}\subset\mathcal{E}^{(j+1)}$ is split (by
$\phi:\mathcal{E}^{(j+1)}\twoheadrightarrow\mathcal{E}^{(i)}$), but then
$\mathcal{F}^{(i)}\subset\mathcal{E}^{(j)}$ is also split (by
$\phi_{|_{\mathcal{E}^{(j)}}}$); that is, $p_{[j]}(q_{\mathcal{E}})\in
Z^{[i]}_{\min}$. ∎

Let us finally relate the non-degeneracy assumption (3) to ‘non-degenerate’ HN
types.

######  Definition 4.38.

A HN type $\tau$ is _non-degenerate_ if there exists a sheaf $\mathcal{E}$ of
HN type $\tau$ such that the inclusion
$\mathcal{E}^{(i-1)}\hookrightarrow\mathcal{E}^{(i)}$ in the HN filtration is
non-split for each $i$ and the quotients
$\mathcal{E}_{i}:=\mathcal{E}^{(i-1)}/\mathcal{E}^{(i)}$ are stable for all
$i$.

The next result is an immediate corollary of Proposition 4.16.

######  Corollary 4.39.

For a HN type $\tau$, Assumption (3) of (QiS) holds if and only if $\tau$ is
non-degenerate.

####  4.4.3. The Upstairs Stabiliser Assumption (U)

We can now give the stabiliser conditions ‘upstairs’ on $X$ that will suffice
to allow us to carry out Construction 4.20.

######  Assumption 4.40 (_The Upstairs Stabiliser Assumption_).

Recall that $U^{[i-1,i]}:=U^{[i-1]}\cap U^{[i]}$.

  1. (1)

We say that the _upstairs unipotent stabiliser assumption_ (UU) holds if for
all $i$,

(UU) $\dim\operatorname{Stab}_{U^{[i]}}(-)\>\text{ and
}\>\dim\operatorname{Stab}_{U^{[i-1,i]}}(-)\>\text{ are constant on
}X_{\min}^{[i],ss}.$

  2. (2)

We say that the _upstairs reductive stabiliser assumption_ (UR) holds if for
all $1<i<l$,

(UR) $\dim\operatorname{Stab}_{R^{(i)}}(z)=0\>\text{ for all }z\in
Z_{\min}^{(i),ss}.$

We say the _Upstairs Stabiliser Assumption_ (U) holds if both the upstairs
unipotent and reductive stabiliser assumptions hold.

######  Remark 4.41.

The fact that we have to fix the dimension of stabilisers for both $U^{[i]}$
and $U^{[i-1,i]}$ is eventually explained by Proposition 5.13; in fact, if $X$
is connected, it is equivalent to fixing their difference by semi-continuity
of the dimension of stabiliser groups.

We will eventually consider the case when (U) fails and perform blow-up
sequence for reductive and unipotent stabilisers. For the unipotent
stabilisers we will use §2.3.7 based on [1]. As observed in §2.3.7, as well as
[3] and [25], we can obtain an explicit description of the open set we get a
quotient of, provided the minimal weight space is not entirely blown up at any
stage of the process. To ensure this, we introduce the following weaker
assumption than (U).

######  Assumption 4.42 (_The Weak Unipotent Upstairs Stabiliser Assumption_).

If $X^{P-qs}\neq\emptyset$ and there exists $z\in Z_{\min}$ having minimal
dimensional stabilisers for the groups $U^{[i]}$ and $U^{[i-1,i]}$ i.e.

[$Z_{\min}\not\subseteq C$]
$\dim\operatorname{Stab}_{U^{[i]}}(z)=d^{[i]}_{\min}\quad\text{and}\quad\dim\operatorname{Stab}_{U^{[i-1,i]}}(z)=d^{[i-1,i]}_{\min}\quad\text{
for all }i,$

we say that the _weak upstairs unipotent stabiliser assumption_ (WUU) holds.

######  Remark 4.43.

Note that if (WUU) holds, then the set of $z\in Z_{\min}$ having minimal
dimensional stabilisers for $U^{[i]}$ and $U^{[i-1,i]}$ is non-empty and open.
If $X$ is irreducible, so is $Z_{\min}$ and $Z^{s}_{\min}\subset Z_{\min}$ is
a dense open, which intersects the open locus where the stabilisers for
$U^{[i]}$ and $U^{[i-1,i]}$ are minimal, because we assumed
$X^{P-qs}\neq\emptyset$.

####  4.4.4. The Upstairs Stabiliser Assumption for Sheaves

Let us return to our running example of sheaves of fixed HN type $\tau$. By
Lemma 4.15, the Upstairs Reductive Assumption (UR) holds precisely when $\tau$
is a coprime HN type (semistability coincides with stability for each HN-
subquotient). Unfortunately, while the quotienting-in-stages process tackles
the problem with the central torus, the situation for the unipotent
stabilisers is not so good. Often both (UU) and (WUU) fail to hold for
sheaves. However, they are more readily verified for Higgs sheaves (and via
the spectral correspondence Higgs sheaves correspond to certain sheaves on a
higher dimensional base): one can show (WUU) holds if one works with a Higgs
HN type whose underlying sheaf HN type is non-empty [18].

Unipotent stabilisers of sheaves (and more general moduli problems in a linear
abelian category) are better behaved with respect to the quotienting-in-stages
framework as follows.

######  Lemma 4.44.

For all $q_{\mathcal{E}}\in X_{\min}^{ss}=Y_{\tau}^{ss}$ and all $i$, the
dimensions of the $U_{\tau}^{[i]}$-stabilisers (resp.
$U_{\tau}^{[i-1,i]}$-stabilisers) of $q_{\mathcal{E}}$ and
$p_{[i]}(q_{\mathcal{E}})$ coincide.

###### Proof.

This follows from Proposition 4.17. ∎

######  Remark 4.45.

If (QiS) and (UU) hold, we claim that the Quotienting-in-Stages stable locus
$X^{P-qs}$ is open and non-empty. Indeed, as
$\dim\operatorname{Stab}_{U^{[i]}}(-)$ is constant on $X_{\min}^{[i],ss}$,
then $U^{[i]}Z^{[i],(s)s}_{\min}\subset X_{\min}^{[i],(s)s}$ is a closed
subset by [1, Lemma 5.4]. Note that $X^{(s)s}_{\min}\subset
X_{\min}^{[i],(s)s}$ by Lemma 4.33 (v). Hence, $X^{P-qs}$ is open in
$X^{s}_{\min}$, which itself is open in $X_{\min}$. Thus provided $X_{\min}$
is open in $X$, which for example will be the case if $X$ is
irreducible121212See $\S$5.4 for how to remove this irreducibility
assumption., the Quotienting-in-Stages stable locus $X^{P-qs}$ is open in $X$.
Finally, non-degeneracy (3) of (QiS) implies this set is non-empty.

###  4.5. Roadmap of the proof of Theorem 1.2

In the next section, we will prove our main result, Theorem 1.2, on this
quotienting-in-stages procedure. Let us first outline the proof.

In Corollary 4.26, we saw that if (D) holds then we obtain a projective
geometric $P$-quotient of some open subset $\operatorname{dom}(q)\subset X$.
In §5.1, we first show under assumption (D) that the domain of definition of
this quotient map $q$ has an explicit description (see Theorem 5.1). This
proof is a form of induction on the length $l$, making use of the auxiliary
conditions $D(i)$, $E(i)$, $F(i)$, which are defined below in Definition 5.2.
The basic idea is that, as a consequence of (QiS), the stage $i$ minimal
weight space $Z_{i,\min}$ downstairs in $X_{i}$ is the image under $q_{(i-1)}$
of the $[i]$-minimal weight space upstairs (see Lemma 5.4). After this, the
most subtle point in the proof of Theorem 5.1 lies in ‘pushing forward’
reductive GIT stability of points in minimal weight spaces along the quotient
maps $q_{(i)}$ (see Proposition 5.7).

The next step is to replace (D) with (U) and in this case show that the domain
of $q$ coincides with the Quotienting-in-Stages stable locus $X^{P-qs}$
introduced in Definition 4.11. For this, we prove some comparison results for
upstairs versus downstairs unipotent stabiliser dimensions in §5.2.1, which
allow us to deduce Proposition 5.14 that gives the part of Theorem 1.2 under
the assumptions (QiS) and (U): namely $X^{P-qs}=\operatorname{dom}(q)$ is a
non-empty open, which admits a projective geometric $P$-quotient.

In §5.2.3, we turn our attention to what happens when (QiS) and (UU) hold, but
(UR) fails. In this case we much perform partial desingularisations as in [28]
for the reductive groups $R^{(i)}$ until we achieve (UR) on the blown up space
$\widehat{X}$, whilst also preserving (QiS) and (UU) (see Proposition 5.20).
Then, in Proposition 5.21 we simply apply Proposition 5.14 to $\widehat{X}$ to
deal with this case. Finally in $\S$5.3 we deal with the case when (UU) fails
by performing a blow-up sequence concerning unipotent stabilisers as described
in $\S$2.3.7, which is based on an earlier arXiv version of [1]. To achieve
our result, we must avoid the bad situation where the whole minimal weight
space is blown up at any stage, and this is where we use the assumption (WUU)
to complete the proof of Theorem 1.2.

##  5\. Quotienting-in-stages for parabolics: Proof of the Main Theorem

In this section, we assume that we have a linearised action of a parabolic
group $P<\mathrm{SL}_{n}$ on an irreducible projective scheme $X$ as in
$\S$4.2. We now proceed with the proof of Theorem 1.2, as outlined in $\S$4.5
above. Throughout this section, we assume that the Quotienting-in-Stages
Assumption (QiS) holds.

###  5.1. Describing the stable locus under the Downstairs Stabiliser
Assumption (D)

As a first step in our proof of Theorem 1.2, we will obtain the following
result.

######  Theorem 5.1.

Suppose that (QiS) and (D) hold. Then we have

$\operatorname{dom}(q^{s})=\operatorname{dom}(q)=\left\\{x\in
X^{s}_{\min}\>\middle|\>p_{[j]}(x)\notin U^{[j-1]}Z^{[j-1]}_{\min}\text{ for
all }2\leq j\leq l\right\\}$

and this locus has a projective geometric $P$-quotient.

Recall that by Corollary 4.28 we know that when (QiS) and (D) hold we have
$\operatorname{dom}(q)\subset X^{s}_{\min}$, so that $q=q^{s}$ and to prove
the above theorem it suffices to prove $\operatorname{dom}(q^{s})$ is the
explicit set described above.

Our proof of Theorem 5.1 is inductive, making use of the following conditions.

######  Definition 5.2.

(Description of $D(i),E(i)$ and $F(i)$) Let $1\leq i\leq l-1$.

  1. (1)

We say that the _$i$ th Description holds_ (denoted $D(i)$) if

$\operatorname{dom}(q^{s}_{(i)})\cap Z^{[i+1]}_{\min}\neq\emptyset.$

We write $D(\leq k)$ if the above holds for all $i\leq k$.

  2. (2)

We say that the _$i$ th Explicit description holds_ (denoted $E(i)$) if

$\operatorname{dom}(q^{s}_{(i)})=\mathcal{E}(i):=\left\\{x\in
X^{s}_{\min}\setminus
U^{[i]}Z^{[i],s}_{\min}\>\middle|\>p_{[j]}(x)\in\operatorname{dom}(q_{(j-1)})\text{
for all }2\leq j\leq i\right\\}$

We say $E(\leq k)$ holds if $E(i)$ holds for all $i\leq k$.

  3. (3)

We say that the _$i$ th Final description holds_ (denoted $F(i)$) if

$\operatorname{dom}(q^{s}_{(i)})=\mathcal{F}(i):=\left\\{x\in
X^{s}_{\min}\setminus U^{[i]}Z^{[i],s}_{\min}\>\middle|\>p_{[j]}(x)\notin
U^{[j-1]}Z^{[j-1]}_{\min}\text{ for all }2\leq j\leq i\right\\}.$

We say $F(\leq k)$ holds if $F(i)$ holds for all $i\leq k$.

In particular, for $i=l-1$, we get

$\mathcal{F}(l-1)=\left\\{x\in X^{s}_{\min}\>\middle|\>p_{[j]}(x)\notin
U^{[j-1]}Z^{[j-1]}_{\min}\text{ for all }2\leq j\leq l\right\\}$

since $\lambda^{[l]}$ is trivial, thus $p_{[l]}=\mathrm{Id}_{X}$ (see Remark
4.10). Thus in Theorem 5.1, we will show that
$\operatorname{dom}(q^{s})=\mathcal{F}(l-1)$. In fact, under the upstairs
stabiliser assumptions (U), one has $\mathcal{F}(l-1)=X^{P-qs}$.

######  Remark 5.3.

In addition to the above (set-theoretic) descriptions, we can characterise the
sets $\mathcal{F}(i)$ and $\mathcal{E}(i)$ scheme-theoretically using the
equalities:

$\mathcal{E}(i):=\left(X^{s}_{\min}\setminus
U^{[i]}Z^{[i],s}_{\min}\right)\cap\left(\bigcap_{2\leq j\leq
i}p_{[j]}^{-1}(\operatorname{dom}(q_{(j-1)})\cap Z^{[j]}_{\min})\right)$
$\mathcal{F}(i):=X^{s}_{\min}\setminus\left(U^{[i]}Z^{[i],s}_{\min}\cup\bigcup_{2\leq
j\leq i}p_{[j]}^{-1}(U^{[j-1]}Z^{[j-1]}_{\min}\cap Z^{[j]}_{\min})\right).$

We now indicate the strategy of our proof. We will inductively show that
$F(i)$ holds, by proving the following implications

(12) $\begin{array}[]{l}D(\leq i-1)\\\ E(\leq i-1)\\\ F(\leq
i-1)\end{array}\implies\begin{array}[]{l}D(\leq i-1)\\\ E(\leq i)\\\ F(\leq
i-1)\end{array}\implies\begin{array}[]{l}D(\leq i-1)\\\ E(\leq i)\\\ F(\leq
i)\end{array}\implies\begin{array}[]{l}D(\leq i)\\\ E(\leq i)\\\ F(\leq
i)\end{array}$

For $i=1$, under the assumption $[\widehat{U}_{1};R_{1}\text{-ss}]$ on
$X_{1}=X$, we have by definition of $q^{s}_{(1)}$ that

$\operatorname{dom}(q^{s}_{(1)})=X^{s}_{\min}\setminus
U^{[1]}Z^{[1],s}_{\min}$

and, in particular, $E(1)$ and $F(1)$ hold, which form the base step of our
inductive proof.

Before we begin the induction, let us note some immediate consequences of
condition $D(i)$.

####  5.1.1. Consequences of $D(i)$

######  Lemma 5.4.

If $D(i)$ holds, then the following hold for the map $q_{(i)}:X\dashrightarrow
X_{i+1}$.

  1. _(1)_

For $x\in q_{(i)}^{-1}(Z_{i+1,\min})$, we have
$p_{[i+1]}(x)\in\operatorname{dom}(q_{(i)})$,

  2. _(2)_

For $x\in\operatorname{dom}(q_{(i)})$, we have $q_{(i)}(x)\in X_{i+1,\min}$ if
and only if $x\in X^{[i+1]}_{\min}$ and also
$p_{[i+1]}(x)\in\operatorname{dom}(q_{(i)})$,

  3. _(3)_

We have
$q_{(i)}^{-1}(Z_{i+1,\min})=(U^{[i]}Z^{[i+1]}_{\min})\cap\operatorname{dom}(q_{(i)})$,

  4. _(4)_

In particular, for all $x\in\operatorname{dom}(q_{(i+1)})$, we have
$p_{[i+1]}(x)\in\operatorname{dom}(q_{(i)})$.

###### Proof.

As a conequence of $D(i)$, we have

(13) $\operatorname{dom}(q_{(i)})\cap Z^{[i+1]}_{\min}\neq\emptyset.$

The map $q_{(i)}$ is a $\widehat{H}^{(i)}$-quotient which is constructed by
taking invariants and so looks like a projection
$\mathbb{P}(V^{*})\dashrightarrow\mathbb{P}(W^{*})$ where
$V=H^{0}(X,\mathcal{L}^{\otimes N})\supset W:=V^{\widehat{H}^{(i)}}$ and $N$
is sufficiently large. Then Equation (13) implies that the maximal weights for
$\lambda^{[i+1]}(\mathbb{G}_{m})$ acting on both $V$ and $W$ coincide. Hence,
we can choose a basis of $V$ of the form

$V=<\alpha_{1},\dots,\alpha_{n},\alpha_{1}^{\prime},\dots,\alpha^{\prime}_{n^{\prime}},\beta_{1},\dots\beta_{m},\beta^{\prime}_{1},\dots,\beta^{\prime}_{m^{\prime}}>$

where
$W=<\alpha_{1},\dots,\alpha_{n},\alpha_{1}^{\prime},\dots,\alpha^{\prime}_{n^{\prime}}>$
and the maximal weight space for $\lambda^{[i+1]}(\mathbb{G}_{m})$ in $V$ is
spanned by $\alpha_{1},\dots,\alpha_{n},\beta_{1},\dots\beta_{m}$. With
respect to this basis, the projection
$\mathbb{P}(V^{*})\dashrightarrow\mathbb{P}(W^{*})$ corresponding to $q_{(i)}$
is the projection onto the first $n+n^{\prime}$ coordinates (corresponding to
the elements $\alpha_{j}$ and $\alpha^{\prime}_{j}$). Moreover, the projection
$p_{[i+1]}:X\dashrightarrow Z^{[i+1]}_{\min}$ to the minimal weight space is
the projection onto the coordinates corresponding to $\alpha_{j}$ and
$\beta_{j}$. Thus a point lies in $X^{[i+1]}_{\min}$ precisely when one of
these coordinates is non-zero.

For (1), we use the above descriptions of $q_{(i)}$ and $p_{[i+1]}$. Let
$x\in\operatorname{dom}(q_{(i)})$ with $q_{(i)}(x)\in Z_{i+1,\min}$; then all
the $\alpha^{\prime}_{j}$ coordinates of $x$ are zero. However, as
$x\in\operatorname{dom}(q_{(i)})$, there must be one $\alpha_{j}$ coordinate
of $x$ that is non-zero, which also means that $p_{[i+1]}(x)$ has a non-zero
$\alpha_{j}$ coordinate and thus is in the domain of definition of $q_{(i)}$.

For (2), we also use these descriptions of $q_{(i)}$ and $p_{[i+1]}$. Since
$x\in\operatorname{dom}(q_{(i)})$, one of the $\alpha_{j}$ or
$\alpha^{\prime}_{j}$ coordinates of $x$ is non-zero. Then
$p_{i+1}(q_{(i)}(x))$ is the projection onto the $\alpha_{j}$ coordinates of
$x$ and so $q_{(i)}(x)\in X_{i+1,\min}$ if and only if one of the $\alpha_{j}$
coordinates of $x$ is non-zero. Similarly $x\in X^{[i+1]}_{\min}$ and
$p_{[i+1]}(x)\in\operatorname{dom}(q_{(i)})$ is equivalent to one of the
$\alpha_{j}$ coordinates of $x$ being non-zero.

For (3), let us first show the inclusion of the right side in the left. If
$x\in\operatorname{dom}(q_{(i)})\cap U^{[i]}Z^{[i+1]}_{\min}$, then we claim
that $q_{(i)}(x)\in Z_{i+1,\min}$. Since $q_{(i)}$ is $U^{(i)}$-invariant, we
can assume $x\in Z^{[i+1]}_{\min}$; then its image $z:=q_{(i)}(x)\in X_{i+1}$
is fixed by the 1-PS $\lambda_{i+1}$ induced by the $\lambda^{[i+1]}$-action
on $X$. Furthermore, as the induced linearisation
$\mathcal{L}_{i+1}\rightarrow X_{i+1}$ satisfies
$q_{(i)}^{*}\mathcal{L}_{i+1}\cong\mathcal{L}^{\otimes N}$, the
$\lambda_{i+1}$-weight of $\mathcal{L}_{i+1}$ over $z$ equals the
$\lambda^{[i+1]}$-weight of $\mathcal{L}^{\otimes N}$ over $x$, as both of
these points are fixed by these 1-PSs and $z=q_{(i)}(x)$. By Equation (13),
the minimal weight for $\lambda_{i+1}$ acting on $\mathcal{L}_{i+1}\rightarrow
X_{i+1}$ is the minimal weight as for $\lambda^{[i+1]}$ acting on
$\mathcal{L}^{\otimes N}\rightarrow X$. Thus the $\lambda^{[i+1]}$-fixed point
$x$ has minimal $\lambda^{[i+1]}$-weight if and only if the
$\lambda_{i}$-fixed point $z$ has minimal $\lambda_{i+1}$-weight. Therefore,
$z\in Z_{i+1,\min}$ as required. For the inclusion of the left side in the
right, suppose that $x\in\operatorname{dom}(q_{(i)})$ and $z=q_{(i)}(x)\in
Z_{i+1,\min}$. By (1), we know that
$\overline{x}:=p_{[i+1]}(x)\in\operatorname{dom}(q_{(i)})$ and also
$q_{(i)}(\overline{x})=p_{i+1}(q_{(i)}(x))=z$. Arguing as above, the
$\lambda^{[i+1]}$-weight of $\overline{x}$ agrees with the
$\lambda_{i+1}$-weight of $z$, which is minimal and thus $\overline{x}\in
Z^{[i+1]}_{\min}$. Since $q_{(i)}(\overline{x})=z=q_{(i)}(x)$ and $q_{(i)}$ is
a geometric $\widehat{H}^{(i)}$-quotient, $x$ lies in the
$\widehat{H}^{(i)}$-orbit of $\overline{x}\in Z^{[i+1]}_{\min}$. Since
$\lambda^{[i+1]}$ is central in the Levi $L$, the action of the Levi preserves
$Z^{[i+1]}_{\min}$, thus
$x\in\widehat{H}^{(i)}Z^{[i+1]}_{\min}=U^{[i]}Z^{[i+1]}_{\min}$ as required.

For (4), since
$x\in\operatorname{dom}(q_{(i+1)})=q_{(i)}^{-1}(\operatorname{dom}(q_{i+1}))$,
we have $q_{(i)}(x)\in\operatorname{dom}(q_{i+1})\subset X_{i+1,\min}$ and so
we conclude that $p_{[i+1]}(x)\in\operatorname{dom}(q_{(i)})$ using (2). ∎

Lemma 5.4 tells us that, provided $D(i)$ holds, the image of
$Z^{[i+1]}_{\min}$ under the quotient map $q_{(i)}$ will be the new minimal
weight space $Z_{i+1\min}$. It is then natural to suppose that the same might
be true for the relevant stable loci. This is a more subtle matter, but we
shall confirm this is the case in Proposition 5.7 below; first we need a
technical preliminary lemma.

######  Lemma 5.5.

Suppose that (QiS) and (D) hold. Fix $1<i<l$ and suppose that $D(\leq i-1)$
holds. For $1<j\leq i$ and $x\in\operatorname{dom}(q_{(j-1)})\cap
X^{(i),s}_{\min}$ with $p_{[j]}(x)\in\operatorname{dom}(q_{(j-1)})$, the image
$q_{(j-1)}(p_{[j]}(x))\in Z_{j,\min}$ is stable for the action of
$R^{(j,i)}:=R^{(i)}/R^{(j-1)}$.

###### Proof.

We prove the claim by induction on $j$. For $j=2$, suppose
$x\in\operatorname{dom}(q_{(1)})\cap X^{(i),s}_{\min}$ with
$\overline{x}:=p_{[2]}(x)\in\operatorname{dom}(q_{(1)})$; then we need to show
that $q_{(1)}(\overline{x})\in Z_{2,\min}$ is stable for $R^{(2,i)}$. Consider
the action of the larger non-reductive group

$\widehat{H}^{\prime}:=U^{(1)}\rtimes(T^{(1)}\times
R^{(i)})>\widehat{H}^{(1)}=U^{(1)}\rtimes(T^{(1)}\times R^{(1)})$

on $X=X_{1}$, where the unipotent radical $U^{(1)}=U^{[1]}$ is graded by
$\lambda^{[1]}(\mathbb{G}_{m})=T^{(1)}$ with minimal weight space
$Z^{[1]}_{\min}$. Recall that the $\widehat{H}^{(1)}$-quotient of
$q_{(1)}:X_{1}\dashrightarrow X_{2}$ is constructed by first taking a
$U^{(1)}$-quotient of $X^{[1]}_{\min}$ and then taking a residual reductive
GIT quotient of $T^{(1)}\times R^{(1)}$ acting on a projective completion
$Y:=\overline{X^{[1],ss}_{\min}/U^{[1]}}$ described in Remark 2.27. Since (DU)
holds and the $R^{(i)}$-semistable locus $Z^{[1],R^{(i)}-ss}_{\min}$ is
contained in the $R^{(1)}$-semistable locus $Z^{[1],ss}_{\min}$, we can take a
NRGIT quotient of the $\widehat{H}^{\prime}$-action on $X=X_{1}$; this is the
reductiveGIT quotient of $X_{2}$ by $R^{(2,i)}$ or equivalently the reductive
GIT quotient of $Y$ by $T^{(1)}\times R^{(i)}$. Hence, we have the following
commutative diagram of quotients

---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{(1)}=q_{\widehat{H}^{(1)}}}$$\scriptstyle{\quad
q_{U^{(1)}}}$$\scriptstyle{q_{\widehat{H}^{\prime}}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{T^{(1)}\times
R^{(1)}}}$$\scriptstyle{q_{T^{(1)}\times
R^{(i)}}\quad\quad\quad\quad}$$\textstyle{Y/\\!/(T^{(1)}\times
R^{(i)})}$$\textstyle{X_{2}=Y/\\!/(T^{(1)}\times
R^{(1)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{R^{(2,i)}}}$

where the subscript denotes the group by which we have quotiented. To prove
that $q_{(1)}(\overline{x})$ is $R^{(2,i)}$-stable, we want to apply Lemma 5.6
below to the action of the action of the product of $G_{1}:=T^{(1)}\times
R^{(1)}$ and $G_{2}:=R^{(2,i)}$ on $Y$. For this, we first need to verify that
$q_{U^{(1)}}(\overline{x})\in Y$ is stable for $G_{1}\times G_{2}$; by Theorem
2.28 __(i)__ , it suffices to check that $p_{[1]}(\overline{x})\in
Z^{[1],R^{(i)}-s}_{\min}$, as we already know that
$\overline{x}=p_{[2]}(x)\in\operatorname{dom}(q_{(1)})\subset
X^{[1]}_{\min}\setminus U^{(1)}Z^{[1]}_{\min}$. By Assumption (1) of (QiS), we
have that $Z^{(i)}_{\min}\subset Z_{\min}^{[1]}$ and in fact this is a closed
$R^{(i)}$-invariant subscheme; therefore,
$Z^{R^{(i)}-s}_{\min}=Z^{(i)}_{\min}\cap Z_{\min}^{[1],R^{(i)}-s}$. Since
$x\in X^{(i),s}_{\min}$ we have also $\overline{x}=p_{[2]}(x)\in
X^{(i),s}_{\min}$ by Remark 4.35 (v); thus $p_{(i)}(\overline{x})\in
Z^{s}_{\min}\subset Z_{\min}^{R^{(i)}-s}$. Since $p_{(i)}\circ
p_{[1]}=p_{(i)}$ on $X^{(i)}_{\min}$ by Remark 4.35 (ii), we deduce that
$p_{[1]}(\overline{x})\in Z^{[1],R^{(i)}-s}_{\min}$, which completes the proof
of the base case $j=2$.

For the inductive step, fix $k<i$ and suppose the claimed statement holds for
all $j\leq k$. To prove the statement for $k+1$, we take
$x\in\operatorname{dom}(q_{(k)})\cap X^{(i),s}_{\min}$ such that
$\overline{x}:=p_{[k+1]}(x)\in\operatorname{dom}(q_{(k)})$ and want to show
that $q_{(k)}(\overline{x})\in Z_{k+1,\min}$ is stable for the action of
$R^{(k+1,i)}$. We have $q_{(k)}=q_{k}\circ q_{(k-1)}$, where
$q_{k}:X_{k}\dashrightarrow X_{k+1}$ is the $k$th stage quotient. Let
$y:=q_{(k-1)}(\overline{x})$; then $y\in\operatorname{dom}(q_{k})\subset
X_{k,\min}\setminus U_{k}Z_{k,\min}$, as
$\overline{x}\in\operatorname{dom}(q_{(k)})=q_{(k-1)}^{-1}(\operatorname{dom}(q_{k}))$.
We want to apply the inductive hypothesis to
$\overline{x}\in\operatorname{dom}(q_{(k-1)})$; for this, note that $x\in
X^{(i),s}_{\min}$ implies also $\overline{x}\in X^{(i),s}_{\min}$ by Remark
4.35 (v), and also $p_{[k]}(\overline{x})\in\operatorname{dom}(q_{(k-1)})$, as
$y:=q_{(k-1)}(\overline{x})\in X_{k,\min}$ and so we can apply Lemma 5.4 (2)
since $D(k-1)$ holds. By the inductive hypothesis applied to
$\overline{x}\in\operatorname{dom}(q_{(k-1)})\cap X^{(i),s}_{\min}$, we know
that $q_{(k-1)}(p_{[k]}(\overline{x}))\in Z_{k,\min}$ is $R^{(k,i)}$-stable.
Therefore

$p_{k}(y)=p_{k}(q_{(k-1)}(\overline{x}))=q_{(k-1)}(p_{[k]}(\overline{x}))$

is $R^{(k,i)}$-stable. It remains to show that
$q_{k}(y)=q_{(k)}(\overline{x})\in Z_{k+1,\min}$ is $R^{(k+1,i)}$-stable. For
this, let $G_{1}:=\lambda_{k}(\mathbb{G}_{m})\times R_{k}$ and
$G_{2}:=R^{(k+1,i)}$; then, as in the base case, we consider the following
composition of quotients

$\textstyle{X_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{U_{k}}\quad\quad}$$\textstyle{Y_{k}:=\overline{X^{ss}_{k,\min}/U_{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\quad
q_{G_{1}}}$$\textstyle{X_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{G_{2}}\quad\quad}$$\textstyle{X_{k+1}/\\!/G_{2}}$

where the subscript on each quotient denotes the quotienting group and $q_{k}$
is the composition of the first two of these rational maps. The resulting
quotient on the right is the NRGIT quotient of $X_{k}$ by
$U_{k}\rtimes(G_{1}\times G_{2})$ (using $\lambda_{k}$ to grade $U_{k}$) and
also the reductive GIT quotient of $Y_{k}$ by $G_{1}\times G_{2}$. To deduce
that $q_{k}(y)=q_{G_{1}}(q_{U_{k}}(y))$ is stable with respect to
$G_{2}=R^{(k+1,i)}$, we want to apply Lemma 5.6 to the action of $G_{1}\times
G_{2}$ on $q_{U_{k}}(y)\in Y_{k}$; for this, it remains to check that
$q_{U_{k}}(y)$ is stable for the product $G_{1}\times G_{2}$. Since we have
shown $y\in X_{k,\min}\setminus U_{k}Z_{k,\min}$ and $p_{k}(y)$ is
$R^{(k,i)}$-stable, it follows that $q_{U_{k}}(y)$ is $(G_{1}\times
G_{2})$-stable by Theorem 2.28 __(i)__ applied to the NRGIT quotient of
$X_{k}$ by $U_{k}\rtimes(T_{k}\times R^{(k,i)})$. ∎

To complete the proof of this lemma, we need to prove the following lemma on
performing reductive GIT quotients in stages.

######  Lemma 5.6.

Let $G=G_{1}\times G_{2}$ be a product of reductive groups acting linearly on
a projective scheme $X$ with reductive GIT quotient $q_{G}:X\dashrightarrow
X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}G$.
If $x\in X$ is $G$-stable, then $q_{G_{1}}(x)$ is $G_{2}$-stable, where
$q_{G_{1}}:X\dashrightarrow
X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}G_{1}$
is the $G_{1}$-quotient.

###### Proof.

Let $y:=q_{G_{1}}(x)\in
Y:=X\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}G_{1}$
and write $q_{G_{2}}:Y\dashrightarrow
Y\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}G_{2}$;
then $y\in Y^{G_{2}-ss}=\operatorname{dom}(q_{G_{2}})$ as $x\in
X^{G-ss}=\operatorname{dom}(q_{G})$ and $q_{G}=q_{G_{2}}\circ q_{G_{1}}$. To
prove that $y$ is $G_{2}$-stable, it suffices to show that its orbit is closed
in the semistable locus and its stabiliser is zero dimensional.

Suppose that there is an orbit $G_{2}\cdot y^{\prime}$ of
$y^{\prime}=q_{G_{1}}(x^{\prime})\in Y^{G_{2}-ss}$ contained in the closure of
$G_{2}\cdot y$. Then $q_{G_{2}}(y)=q_{G_{2}}(y^{\prime})$ and also
$q_{G}(x)=q_{G}(x^{\prime})$. Since $x$ is $G$-stable, $x$ and $x^{\prime}$
lie in the same $G$-orbit and thus also $y$ and $y^{\prime}$ lie in the same
$G_{2}$-orbit. Hence $G_{2}\cdot y$ is closed in $Y^{G_{2}-ss}$.

It remains to show that $\operatorname{Stab}_{G_{2}}(y)$ is zero dimensional.
If $\dim\operatorname{Stab}_{G_{2}}(y)>0$, then for each
$s\in\operatorname{Stab}_{G_{2}}(y)$ there is $g_{s}\in G_{1}$ such that
$g_{s}sx=x$, as the $G$-stable point $x$ is also $G_{1}$-stable (for example,
this follows by the Hilbert-Mumford criterion) and so
$q_{G_{1}}^{-1}(y)=G_{1}\cdot x$. The projection of the set

$S:=\\{g_{s}\cdot s\mid
s\in\operatorname{Stab}_{G_{2}}(y)\\}\subseteq\operatorname{Stab}_{G_{1}\times
G_{2}}(x)\leq G_{1}\times G_{2}$

to $G_{2}$ is $\operatorname{Stab}_{G_{2}}(y)$; thus $S$ also has strictly
positive dimension, which contradicts $x\in X^{G-s}$. ∎

We can now show, when $D(\leq i-1)$ holds, that the image of stable
$[i]$-minimal weight space $Z^{[i],s}_{\min}$ under the quotient map
$q_{(i-1)}$ is the stage $i$ stable minimal weight space $Z_{i,\min}^{s}$.

######  Proposition 5.7.

Assume that (QiS) and (D) hold. If $D(\leq i-1)$ holds, then for all
$x\in\operatorname{dom}(q_{(i-1)})\cap Z^{[i],s}_{\min}\cap X^{(i),s}_{\min}$
we have $q_{(i-1)}(x)\in Z_{i,\min}^{s}$.

###### Proof.

Our goal is to use Lemma 5.5 with $j=i$. This applies as for
$x\in\operatorname{dom}(q_{(i-1)})\cap Z^{[i],s}_{\min}\cap X^{(i),s}_{\min}$,
we have $p_{[i]}(x)=x\in\operatorname{dom}(q_{(i-1)})$. We conclude that
$q_{(i-1)}(p_{[i]}(x))$ is stable for the action of $R_{i}$. Since
$p_{[i]}(x)=x$, this completes the proof that $q_{(i-1)}(x)\in
Z_{i,\min}^{s}$. ∎

####  5.1.2. The induction

We now begin our induction. Throughout this section, we assume that (QiS) and
(D) hold.

######  Lemma 5.8.

If $F(\leq i)$ holds, then $D(i)$ holds.

###### Proof.

By the non-degeneracy assumption (3) of (QiS), there exists $x\in
X^{s}_{\min}$ with $p_{[j]}(x)\notin U^{[j-1]}Z^{[j-1]}_{\min}$ for all
$1<j<l$. Since $X^{s}_{\min}\subset X^{[i+1],s}_{\min}$ by Lemma 4.33 (iv), we
have $y:=p_{[i+1]}(x)\in Z^{[i+1]}_{\min}$. Therefore, it suffices to show
that also $y\in\mathcal{F}(i)$, since we have
$\mathcal{F}(i)=\operatorname{dom}(q^{s}_{(i)})$ as $F(i)$ holds. By
assumption, we have $y=p_{[i+1]}(x)\notin U^{[i]}Z^{[i]}_{\min}$. As $x\in
X^{s}_{\min}$, we have by $y=p_{[i+1]}(x)\in X^{s}_{\min}$ by Lemma 4.33 (v).
Thus it remains to check that $p_{[j]}(y)\notin U^{[j-1]}Z^{[j-1]}_{\min}$ for
all $j\leq i$. However, if $p_{[j]}(y)\in U^{[j-1]}Z^{[j-1]}_{\min}$, then as
$p_{[j]}\circ p_{[i+1]}=p_{[i+1]}\circ p_{[j]}$ on $X_{\min}$ by Remark 4.35
(ii), we would obtain by Assumption (2) of (QiS) that $p_{[j]}(x)\in
U^{[j-1]}Z^{[j-1]}_{\min}$, which contradicts our assumption on $x$.
Therefore, $p_{[j]}(y)\notin U^{[j-1]}Z^{[j-1]}_{\min}$ and we have shown
$y\in\mathcal{F}(i)$ as required. ∎

We next prove the middle implication in (12).

######  Proposition 5.9.

If $E(\leq i)$ and $F(\leq i-1)$ hold, then $F(i)$ also holds; that is,

(14) $\operatorname{dom}(q^{s}_{(i)})=\mathcal{F}(i):=\left\\{x\in
X^{s}_{\min}\setminus U^{[i]}Z^{[i],s}_{\min}\>\middle|\>p_{[j]}(x)\notin
U^{[j-1]}Z^{[j-1]}_{\min}\text{ for all }2\leq j\leq i\right\\}.$

###### Proof.

Suppose that $x\in\operatorname{dom}(q^{s}_{(i)})$. Since $E(i)$ holds, we
have $x\in\mathcal{E}(i)$ and in particular, $x\in X^{s}_{\min}$ and $x\notin
U^{[i]}Z^{[i]}_{\min}$. Thus to show $x\in\mathcal{F}(i)$, it remains to check
that $p_{[j]}(x)\notin U^{[j-1]}Z^{[j-1]}_{\min}$ for all $j\leq i$. We have
$\operatorname{dom}(q^{s}_{(i)})\subset\operatorname{dom}(q^{s}_{(i-1)})=\mathcal{F}(i-1)$
as $F(i-1)$ holds; thus $p_{[j]}(x)\notin U^{[j-1]}Z^{[j-1]}_{\min}$ for all
$j\leq i-1$. If $p_{[i]}(x)\in U^{[i-1]}Z^{[i-1]}_{\min}$, then
$p_{[i]}(x)\notin\mathcal{F}(i-1)=\operatorname{dom}(q^{s}_{(i-1)})$ and this
contradicts the fact that $x\in\mathcal{E}(i)$. Therefore, we conclude
$x\in\mathcal{F}(i)$ as required.

Suppose conversely that $x\in\mathcal{F}(i)$. We claim that $x\notin
U^{[i-1]}Z^{[i-1]}_{\min}$. If $x\in U^{[i-1]}Z^{[i-1]}_{\min}$, then there
exists some $u\in U^{[i-1]}$ with $ux\in Z^{[i-1]}_{\min}$ and, as
$Z^{[i-1]}_{\min}$ is closed and preserved by the
$\lambda^{[i]}(\mathbb{G}_{m})$-action, also $p_{[i]}(ux)\in
Z^{[i-1]}_{\min}$. The conjugation action of $\lambda^{[i]}(t)$ on $U^{[i-1]}$
has non-negative weights and so $u_{0}:=\lim_{t\rightarrow
0}\lambda^{[i]}(t)u\lambda^{[i]}(t)^{-1}$ exists in $U^{[i-1]}$. Therefore, we
deduce that $p_{[i]}(ux)=u_{0}p_{[i]}(x)\in Z^{[i-1]}_{\min}$, which
contradicts the assumption that $p_{[i]}(x)\notin U^{[i-1]}Z^{[i-1]}_{\min}$.
Thus $x\notin U^{[i-1]}Z^{[i-1]}_{\min}$ as claimed.

Since by assumption $x\in\mathcal{F}(i)$, we have
$x\in\mathcal{F}(i-1)=\operatorname{dom}(q^{s}_{(i-1)})=\mathcal{E}(i-1)$,
where the equalities follow because $F(i-1)$ and $E(i-1)$ hold. Since $E(i)$
also holds, to show $x\in\operatorname{dom}(q^{s}_{(i)})$ it suffices to show
$x\in\mathcal{E}(i)$. For this, we already know that $x\in\mathcal{E}(i-1)$
and $x\notin U^{[i]}Z^{[i],s}_{\min}$ as $x\in\mathcal{F}(i)$; thus, it
remains to check that
$\overline{x}:=p_{[i]}(x)\in\operatorname{dom}(q_{(i-1)})=\mathcal{F}(i-1)$.
Since $x\in X^{s}_{\min}$, we have $\overline{x}=p_{[i]}(x)\in X^{s}_{\min}$
by Lemma 4.33 (vi). By assumption, $\overline{x}=p_{[i]}(x)\notin
U^{[i-1]}Z_{\min}^{[i-1]}$. If there exists an index $j\leq i-1$, with
$p_{[j]}(\overline{x})\in U^{[j-1]}Z^{[j-1]}_{\min}$, then as
$p_{[j]}(\overline{x})=p_{[i]}(p_{[j]}(x))$ by Lemma 4.33 (iii), we would
deduce $p_{[j]}(x)\in U^{[j-1]}Z^{[j-1]}_{\min}$ from Assumption (2) of (QiS),
which contradicts the fact that
$x\in\operatorname{dom}(q^{s}_{(i-1)})=\mathcal{F}(i-1)$. Therefore,
$\overline{x}\in\mathcal{F}(i-1)=\operatorname{dom}(q^{s}_{(i-1)})$ as
required. ∎

Before we prove the final implication, we prove the following short lemma
relating the $\mathcal{E}(i)$’s.

######  Lemma 5.10.

Suppose that $D(i-2)$ holds. Then for $x\in\mathcal{E}(i)$, we have $x\notin
U^{[i-1]}Z^{[i-1]}_{\min}$. In particular, we have
$\mathcal{E}(i)\subset\mathcal{E}(i-1)$.

###### Proof.

For a contradiction, suppose $x\in U^{[i-1]}Z^{[i-1]}_{\min}$; then we can
conclude $p_{[i]}(x)\in U^{[i-1]}Z^{[i-1]}_{\min}$ exactly as in the second
paragraph in the proof of Proposition 5.9. Since $x\in\mathcal{E}(i)$, we also
know
$p_{[i]}(x)\in\operatorname{dom}(q_{(i-1)})\subset\operatorname{dom}(q_{(i-2)})$.
By Lemma 5.4 $(3)$, we conclude that $q_{(i-2)}(p_{[i]}(x))\in
U_{i-1}Z_{i-1,\min}$. Hence
$q_{(i-2)}(p_{[i]}(x))\notin\operatorname{dom}(q_{i-1})$, which contradicts
$p_{[i]}(x)\in\operatorname{dom}(q_{(i-1)})$. ∎

Finally we prove the left implication in (12).

######  Proposition 5.11.

If $D(\leq i-1)$, $E(\leq i-1)$ and $F(\leq i-1)$ hold, then $E(i)$ holds;
that is,

$\operatorname{dom}(q^{s}_{(i)})=\mathcal{E}(i):=\left\\{x\in
X^{s}_{\min}\setminus
U^{[i]}Z^{[i],s}_{\min}\>\middle|\>p_{[j]}(x)\in\operatorname{dom}(q_{(j-1)})\text{
for all }2\leq j\leq i\right\\}$

###### Proof.

Suppose $x\in\operatorname{dom}(q^{s}_{(i)})$; then $x\in X^{s}_{\min}$ by
definition of $q_{(i)}^{s}$. For all $j\leq i-1$ we have
$\operatorname{dom}(q^{s}_{(i)})\subset\operatorname{dom}(q^{s}_{(j)})=\mathcal{E}(j)$,
where the equality is given by $E(\leq i-1)$. In particular, $x\notin
U^{[j]}Z^{[j]}_{\min}$ and $p_{[j]}(x)\in\operatorname{dom}(q_{(j-1)})$ for
all $j\leq i-1$. Therefore, to show that $x\in\mathcal{E}(i)$, it remains to
check that $x\notin U^{[i]}Z^{[i]}_{\min}$ and
$p_{[i]}(x)\in\operatorname{dom}(q_{(i-1)})$. If $x\in U^{[i]}Z^{[i]}_{\min}$,
then as $D(i-1)$ holds, we have $q_{(i-1)}(x)\in U_{i}Z_{i,\min}$ by Lemma 5.4
(3). Then $q_{(i-1)}(x)\notin\operatorname{dom}(q_{i})\subset
X^{ss}_{i,\min}\setminus U_{i}Z_{i,\min}^{ss}$, where this containment follows
from Theorem 2.28 __(ii)__. However,
$\operatorname{dom}(q_{(i)})=q_{(i-1)}^{-1}(\operatorname{dom}(q_{i}))$ and so
this would contradict the fact that
$x\in\operatorname{dom}(q^{s}_{(i)})\subseteq\operatorname{dom}(q_{(i)})$. If
$p_{[i]}(x)\notin\operatorname{dom}(q_{(i-1)})$, then $q_{(i-1)}(x)\notin
X_{i,\min}$ by Lemma 5.4 (2), which we can apply as $D(i-1)$ holds. This again
contradicts the fact that
$x\in\operatorname{dom}(q_{(i)})=q_{(i-1)}^{-1}(\operatorname{dom}(q_{i}))$,
as $\operatorname{dom}(q_{i})\subset X_{i,\min}$. Therefore, we conclude that
$x\in\mathcal{E}(i)$ as required.

Suppose conversely that $x\in\mathcal{E}(i)$. Then by Lemma 5.10, also
$x\in\mathcal{E}(i-1)=\operatorname{dom}(q^{s}_{(i-1)})$, where this equality
is due to $E(i-1)$. Since $x\in\operatorname{dom}(q^{s}_{(i-1)})$, to prove
that $x\in\operatorname{dom}(q^{s}_{(i)})$, it suffices to show that the point
$y:=q_{(i-1)}(x)\in X_{i}$ is stable for the $\widehat{H}_{i}$-action; that
is, we need to show that $y\in X^{s}_{i,\min}\setminus U_{i}Z_{i,\min}^{s}$.
By assumption, we have $p_{[i]}(x)\in\operatorname{dom}(q_{(i-1)})$ and so
$y=q_{(i-1)}(x)\in X_{i,\min}$ by Lemma 5.4 (2), which we can apply as
$D(i-1)$ holds. Furthermore, $y\notin U_{i}Z_{i,\min}$, as $x\notin
U^{[i-1]}Z^{[i]}_{\min}$ and so this follows by Lemma 5.4 (2), which we can
apply $D(i-1)$ holds. Therefore, it remains to check that $y\in
X^{s}_{i,\min}$ or equivalently $p_{i}(y)\in Z_{i,\min}^{s}$; this last claim
follows from Proposition 5.7, because $p_{i}(y)=q_{(i-1)}(p_{[i]}(x))$ and
$p_{[i]}(x)\in\operatorname{dom}(q_{(i-1)})$ and also $p_{[i]}(x)\in
Z^{[i],s}_{\min}\cap X^{s}_{\min}$ by Lemma 4.33 (v), as $x\in
X^{s}_{\min}\subset X^{[i],s}_{\min}$. ∎

Now we can describe the domain of $q$, under assumptions on the downstairs
stabilisers.

###### Proof of Theorem 5.1.

Since (D) holds, we have by Theorem 2.29 and the definition of $q_{(1)}^{s}$
that

$\operatorname{dom}(q^{s}_{(1)})=X^{s}_{\min}\setminus
U^{[1]}Z^{[1],s}_{\min},$

which shows $E(1)$ and $F(1)$ hold. We prove that the domain of $q^{s}$ is
$\mathcal{F}(l-1)$ inductively using Lemma 5.8, together with Propositions 5.9
and 5.11. Since (D) holds, $\operatorname{dom}(q)\subset X^{s}_{\min}$ by
Corollary 4.28 (see Remark 4.32) and consequently
$\operatorname{dom}(q^{s})=\operatorname{dom}(q)$. Finally this set admits a
projective geometric quotient by Corollary 4.26. ∎

###  5.2. From Downstairs to Upstairs Stabiliser Assumptions

In this subsection, we will prove Theorem 1.2 in the case where (UU) holds
(see Proposition 5.21) by moving from the _Downstairs Stabiliser Assumptions_
(D) to the _Upstairs Stabiliser Assumptions_ (U).

Our proof is split into two cases. In $\S$5.2.2, when (U) holds, we
inductively show (D) holds so we can apply Theorem 5.1. In $\S$5.2.3, where
only (UU) holds, we perform a (reductive) blow-up procedure on $X$ to obtain a
sequence of blow ups $\widehat{X}\rightarrow X$ such that (U) holds on
$\widehat{X}$ and then apply $\S$5.2.2. We deal with the case when (UU) fails
in $\S$5.3 below.

####  5.2.1. Comparing upstairs and downstairs stabilisers

In this subsection, we will prove results relating the assumptions (D) and
(U).

Suppose we have performed Construction 4.20 up to stage $j$, so we have the
rational map

$q_{(j-1)}:X\dashrightarrow X_{j}$

which is equivariant with respect to the quotient map
$\widehat{H}^{(j)}\twoheadrightarrow\widehat{H}_{j}$ and whose restriction to
its domain of definition gives a geometric quotient by

(15) $\widehat{H}^{(j-1)}=U^{(j-1)}\rtimes(R^{(j-1)}\times
T^{(j-1)})\quad\text{where}\quad
T^{(j-1)}=\lambda^{[1]}(\mathbb{G}_{m})\times\dots\times\lambda^{[j-1]}(\mathbb{G}_{m})$

Now we begin the work of relating the upstairs and downstairs stabilisers.

######  Lemma 5.12.

Suppose that (QiS) holds, and that (D) holds up to the $(j-1)$th stage, so
that we can obtain a geometric quotient $q_{(j-1)}$ using Theorem 5.1. Then
for $x\in X^{[j]}_{\min}\cap\operatorname{dom}(q_{(j-1)})$, we have an exact
sequence

(16)
$1\rightarrow\operatorname{Stab}_{U^{[j-1,j]}}(x)\rightarrow\operatorname{Stab}_{U^{[j]}}(x)\rightarrow\operatorname{Stab}_{U_{j}}(q_{(j-1)}(x)).$

If moreover, $x\in Z^{[j]}_{\min}$, then this sequence is also right exact.

###### Proof.

Since $q_{(j-1)}$ is equivariant with respect to the quotient map
$U^{[j]}\twoheadrightarrow U_{j}$, we obtain a natural morphism

$\operatorname{Stab}_{U^{[j]}}(x)\rightarrow\operatorname{Stab}_{U_{j}}(q_{(j-1)}(x))$

whose kernel is $\operatorname{Stab}_{U^{[j]}}(x)\cap
U^{[j-1]}=\operatorname{Stab}_{U^{[j-1,j]}}(x)$, which proves the first claim.

Now suppose additionally $x\in Z^{[j]}_{\min}$ and write $y:=q_{(j-1)}(x)$. We
claim this sequence is also right exact. Let
$\bar{u}\in\operatorname{Stab}_{U_{j}}(y)$; choose a lift $u\in U^{(j)}$ of
$\bar{u}$ under the quotient map $U^{(j)}\twoheadrightarrow U_{j}$. By
equivariance of $q_{(j-1)}$, we have $q_{(j-1)}(ux)=\bar{u}y=y=q_{(j-1)}(x)$,
and as $q_{(j-1)}$ is a geometric $\widehat{H}^{(j-1)}$-quotient, there exists
an element $h\in\widehat{H}^{(j-1)}$ such that $ux=hx$. Let us write $h=vrt$
with respect to the decomposition (15), where $v\in U^{(j-1)}$, $r\in
R^{(j-1)}$ and $t\in T^{(j-1)}$. Since $\lambda^{[j-1]}$ is a central 1-PS of
the Levi $L$ and $R^{(j-1)}\times T^{(j-1)}<L$, the minimal weight space
$Z^{[j]}_{\min}$ is preserved by the action of $R^{(j-1)}\times T^{(j-1)}$.
Thus $v^{-1}ux=rtx\in Z^{[j]}_{\min}$. We claim there is an element
$\tilde{h}\in U^{(j-1)}$ such that $\tilde{u}:=\tilde{h}v^{-1}u\in U^{[j]}$
and $\tilde{u}x\in Z^{[j]}_{\min}$. Indeed if we write the block form of
$v^{-1}u\in U^{(j)}$ with respect to the two-term weight filtration of
$\lambda^{[j]}$: if

$v^{-1}u=\left(\begin{array}[]{cc}A&B\\\
0&\operatorname{id}\end{array}\right),\quad\text{then}\quad\tilde{h}:=\left(\begin{array}[]{cc}A^{-1}&0\\\
0&\operatorname{id}\end{array}\right),$

has the desired properties: $\tilde{h}$ fixes $Z^{[j]}_{\min}$ setwise since
it lies in the Levi $R^{[j]}$ of the parabolic $P^{[j]}=P(\lambda^{[j]})$.
Since $\tilde{u}\in U^{[j]}$ and $\tilde{u}x\in Z^{[j]}_{\min}$, it follows
that $\tilde{u}\in\operatorname{Stab}_{U^{[j]}}(x)$ by Lemma 2.23. Finally,
the image of $\tilde{u}=\tilde{h}v^{-1}u$ under the quotient
$U^{[j]}\twoheadrightarrow U_{j}$ is $\bar{u}$, as $\tilde{h}v^{-1}\in
U^{(j-1)}$, which completes the surjectivity proof. ∎

######  Proposition 5.13.

In the situation of Lemma 5.12, suppose additionally that (UU) holds. Take
$x\in X^{[j],ss}_{\min}\cap\operatorname{dom}(q_{(j-1)})$ such that
$p_{[j]}(x)\in\operatorname{dom}(q_{(j-1)})$ and $q_{(i-1)}(x)\in Y_{j,\min}$.
Then

$\dim\operatorname{Stab}_{U_{j}}(q_{(j-1)}(x))=\dim\operatorname{Stab}_{U^{[j]}}(x)-\dim\operatorname{Stab}_{U^{[j-1,j]}}(x).$

###### Proof.

Let us write $\mkern 1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern
1.5mu:=p_{[j]}(x)$ and $y:=q_{(j-1)}(x)$. Then $q_{(i-1)}(\mkern
1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu)=\mkern
1.5mu\overline{\mkern-1.5muy\mkern-1.5mu}\mkern 1.5mu$ where $\mkern
1.5mu\overline{\mkern-1.5muy\mkern-1.5mu}\mkern 1.5mu=p_{j}(y)$. By applying
Lemma 5.12 to $\mkern 1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu\in
Z^{[j]}_{\min}\cap\operatorname{dom}(q_{(j-1)})$, we have

$\dim\operatorname{Stab}_{U_{j}}(\mkern
1.5mu\overline{\mkern-1.5muy\mkern-1.5mu}\mkern
1.5mu)=\dim\operatorname{Stab}_{U^{[j]}}(\mkern
1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern
1.5mu)-\dim\operatorname{Stab}_{U^{[j-1,j]}}(\mkern
1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu),$

and by applying this lemma also to $x$, we have

(17)
$\dim\operatorname{Stab}_{U_{j}}(y)\geq\dim\operatorname{Stab}_{U^{[j]}}(x)-\dim\operatorname{Stab}_{U^{[j-1,j]}}(x).$

By the semicontinuity of dimensions of stabilisers, we have

$\dim\operatorname{Stab}_{U_{j}}(y)\leq\dim\operatorname{Stab}_{U_{j}}(\mkern
1.5mu\overline{\mkern-1.5muy\mkern-1.5mu}\mkern 1.5mu).$

We claim that equality holds here and in (17): indeed, we have

$\begin{split}\dim\operatorname{Stab}_{U_{j}}(\mkern
1.5mu\overline{\mkern-1.5muy\mkern-1.5mu}\mkern
1.5mu)\geq\dim\operatorname{Stab}_{U_{j}}(y)&\geq\dim\operatorname{Stab}_{U^{[j]}}(x)-\dim\operatorname{Stab}_{U^{[j-1,j]}}(x)\\\
&=\dim\operatorname{Stab}_{U^{[j]}}(\mkern
1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern
1.5mu)-\dim\operatorname{Stab}_{U^{[j-1,j]}}(\mkern
1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu)\\\
&=\dim\operatorname{Stab}_{U_{j}}(\mkern
1.5mu\overline{\mkern-1.5muy\mkern-1.5mu}\mkern 1.5mu),\end{split}$

where the second to last equality follows from (UU) as $x\in
X^{[j],ss}_{\min}$. ∎

####  5.2.2. The case when the Upstairs Stabiliser Assumption holds

In this section, we assume the $P$-action on $X$ satisfies (QiS) and also (U).

######  Proposition 5.14.

If (U) holds for the linearised $P$-action on $X$, then the Quotienting-in-
Stages procedure yields a projective geometric $P$-quotient of the locus

$\operatorname{dom}(q^{s})=\operatorname{dom}(q)=X^{P-qs}.$

###### Proof.

We will inductively show that if (U) holds, then (D) and thus, the result
follows from Theorem 5.1, as when (U) holds we have

$X^{P-qs}=\left\\{x\in X^{s}_{\min}\>\middle|\>p_{[j]}(x)\notin
U^{[j-1]}Z^{[j-1]}_{\min}\text{ for all }2\leq j\leq l\right\\}.$

For the base case, if $l=2$, then there is only one stage in the Quotienting-
in-Stages procedure and the Upstairs and Downstairs Stabiliser Assumptions
(that is, (U) and (D)) coincide. For the inductive step, we fix $i$ and assume
that the conditions in (D) hold for the groups $U_{j}$ and $R_{j}$ for all
$j\leq i$, then

$q_{(i)}:X\dashrightarrow X_{i+1}$

is a geometric $\widehat{H}^{(i)}$-quotient of its domain of definition, which
is $\operatorname{dom}(q_{(i)}^{s})=\mathcal{F}(i)$ following the inductive
proof of Theorem 5.1 up to this stage. We need to show that we can deduce the
stabiliser conditions in (D) for the groups $U_{i+1}$ and $R_{i+1}$ acting on
$X_{i+1}$ by using the Assumption (U) upstairs on $X$. For any $y\in
X^{s}_{i+1,\min}$ we can write $y=q_{(i)}(x)$ for some
$x\in\mathcal{F}(i)\subset X^{s}_{\min}\subset X^{[i],s}_{\min}$. By Lemma 5.8
we can apply Lemma 5.4 (2) to deduce that
$p_{[i+1]}(x)\in\operatorname{dom}(q_{(i)})$. Hence we can apply Proposition
5.13 to the map $q_{(i)}:X\dashrightarrow X_{i+1}$ to obtain an expression for
$\dim\operatorname{Stab}_{U_{i+1}}(y)$ in terms of the dimensions of the
stabilisers of $x$ under $U^{[i+1]}$ and $U^{[i,i+1]}$. Since (U) holds, the
latter dimensions are constant on $X^{[i],s}_{\min}$ and thus we see that
$\dim\operatorname{Stab}_{U_{i+1}}$ is fixed on $X^{s}_{i+1,\min}$; that is
the part of (D) concerning the $U_{i+1}$-stabilisers on $X_{i+1}$ holds.

To prove the part of (D) concerning the stabilisers for the reductive group
$R_{i+1}$ acting on $X_{i+1}$, we need to show
$Z_{i+1,\min}^{s}=Z_{i+1,\min}^{ss}$. By Lemma 5.4 (3), any $z\in
Z_{i+1,\min}^{ss}$ is the image under $q_{(i)}$ of some $w\in
Z^{[i+1]}_{\min}$. Moreover, as $z$ is $R_{i+1}$-semistable, the point $w$ is
$R^{(i+1)}$-semistable by pulling back invariants along $q_{(i)}$ using Lemma
4.34. To prove that $z\in Z_{i+1,\min}^{s}$, we will show first that it
suffices to show $w\in X^{(i+1),ss}_{\min}$, and then show that $w\in
X^{(i+1),ss}_{\min}$.

We claim that if $w\in X^{(i+1),ss}_{\min}$, then $z\in Z_{i+1,\min}^{s}$. In
this case, we have $w\in X^{(i+1),ss}_{\min}=X^{(i+1),s}_{\min}\subset
X^{[i],s}_{\min}$, where the equality follows from Assumption (UR) and the
inclusion follows from Remark 4.35. Since $w\in Z^{[i+1]}_{\min}$, we see that
$w\in\operatorname{dom}(q_{(i)})\cap Z^{[i+1],s}_{\min}\cap
X^{(i+1),s}_{\min}$ and deduce that $z=q_{(i)}(w)\in Z_{i+1,\min}^{s}$ from
Proposition 5.7.

It remains to show that $w\in X^{(i+1),ss}_{\min}$. For this, we will show $w$
lies in the domain of definition for a NRGIT quotient $\phi$ for a larger
group

$\widehat{H}^{\prime}:=U^{(i)}\rtimes(R^{(i+1)}\times
T^{(i+1)})=\widehat{H}^{(i)}\rtimes(R_{i+1}\times\lambda^{[i+1]}(\mathbb{G}_{m}))$

using arguments similar to Lemma 5.5. Then we will show that that
$\operatorname{dom}(\phi)\subset X^{(i),ss}_{\min}$ using arguments similar to
Corollary 4.28. Recall that by our inductive assumption,
$q_{(i)}:X\dashrightarrow X_{i+1}$ is a geometric $\widehat{H}^{(i)}$-quotient
of its domain of definition and $X_{i+1}$ is the projective spectrum of the
$\widehat{H}^{(i)}$-invariants on $X$, as each stage in the quotient satisfies
the relevant downstairs assumptions of (D) up to this stage. We consider the
following composition of quotients

$\phi:X\dashrightarrow X_{i+1}\dashrightarrow
Y:=X_{i+1}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}(R_{i+1}\times\lambda^{[i]}(\mathbb{G}_{m})),$

where the first quotient is $q_{(i)}$ and the second is a reductive GIT
quotient. Then $\phi$ is also a NRGIT quotient by $\widehat{H}^{\prime}$ and
thus $Y$ is the projective spectrum of the $\widehat{H}^{\prime}$-invariants
on $X$. In fact, we will take these quotients with respect to a perturbation
$\mathcal{L}_{i}\rightarrow X$ of the linearisation. Recall that we obtained a
character $\chi$ to twist the borderline linearisation by in Proposition 4.27
from certain choices $\epsilon_{j}$. We let $\chi_{i}$ be the character
obtained from this construction by setting $\epsilon_{j}=0$ for $j\geq i+1$
and let $\mathcal{L}_{i}$ be the twist of the borderline linearisation by this
character. By construction, the weight of $\lambda_{i+1}$ on
$Z_{i+1,\min}\subset X_{i+1}$ is zero with respect to $\mathcal{L}_{i}$, as
the central torus $T$ acts on $Z_{\min}$ with weight zero with respect to the
borderline linearisation. Therefore,

$Z_{i+1,\min}^{ss}:=(Z_{i+1,\min})^{R_{i+1}-ss}=(Z_{i+1,\min})^{(R_{i+1}\times\lambda_{i+1}(\mathbb{G}_{m}))-ss}(\mathcal{L}_{i})\subset
X_{i+1}^{(R_{i+1}\times\lambda_{i+1}(\mathbb{G}_{m}))-ss}(\mathcal{L}_{i}),$

where we also use $\mathcal{L}_{i}$ to denote the induced linearisation on
$X_{i+1}$ and the last inclusion follows as $Z_{i+1,\min}\hookrightarrow
X_{i+1}$ is an equivariant closed embedding. Thus, since $q_{(i)}(w)=z\in
Z_{i+1,\min}^{ss}$ by assumption, we deduce $w\in\operatorname{dom}(\phi)$. To
conclude that $w\in X^{(i+1),ss}_{\min}$, we will show that
$\operatorname{dom}(\phi)\subset X^{(i+1),ss}_{\min}$. By Assumption (1) of
(QiS), the torus $T^{(i+1)}$ acts trivially on $Z^{(i+1)}_{\min}$ and this
torus must act with weight zero on $Z^{(i+1)}_{\min}$ for the borderline
linearisation $\mathcal{L}_{0}$, as $Z_{\min}\subset Z^{(i+1)}_{\min}$. Hence,

$X_{i+1}^{(R_{i+1}\times\lambda_{i+1}(\mathbb{G}_{m}))-ss}(\mathcal{L}_{i})\subset
X_{i+1}^{(R_{i+1}\times\lambda_{i+1}(\mathbb{G}_{m}))-ss}(\mathcal{L}_{0})=p_{(i+1)}^{-1}(Z^{(i+1),ss}_{\min})=X^{(i+1),ss}_{\min}$

where the first inclusion is by the same VGIT argument as Proposition 4.27.
Thus we see in the same way as Corollary 4.28 that
$\operatorname{dom}(\phi)\subset X^{(i),ss}_{\min}$, which concludes our proof
that $z\in Z_{i+1,\min}^{s}$.

By induction we deduce (D) from the (U) and the result then follows from
Theorem 5.1. ∎

####  5.2.3. The case when only Upstairs Unipotent Stabiliser Assumption holds

In this subsection, we assume that the _Upstairs Unipotent Stabiliser
Assumption_ (UU) holds, but the _Upstairs Reductive Stabiliser Assumption_
(UR) does not. Our approach will be to reduce to the case where both (UU) and
(UR) hold by performing an equivariant blow-up sequence which involves
considering dimensions of stabilisers for the action of the reductive group
$R$ on $X$ analogous to the reductive partial desingularisation procedure [28]
outlined in $\S$2.1.1.

Since we will need to consider different minimal weight spaces on successive
blow-ups of $X$, we introduce the following notation.

######  Definition 5.15.

For any projective scheme $Y$ with an ample linearised $P$-action, we write
$Z_{\min}^{[i]}(Y):=Z_{\min}(Y,\lambda^{[i]})$ for the $\lambda^{[i]}$-minimal
weight space in $Y$ and let $Z_{\min}^{[i],(s)s}(Y)$ denote the
$R^{(i)}$-(semi)stable locus. We write $p_{[i],Y}:Y_{\min}^{[i]}\rightarrow
Z_{\min}^{[i]}(Y)$ for the retraction onto the minimal weight space (or
sometimes just $p_{[i]}$ if $Y$ is fixed). Similarly, we write
$Z_{\min}^{(i),(ss)}(Y)$ for the semistable $\lambda^{(}i)$-minimal weight
space and $Y_{\min}^{(i),(ss)}$ for the corresponding attracting open.

Our goal is to perform sequences of blow-ups for each reductive group
$R^{(i)}$ inductively to arrange that $Z^{(i),s}_{\min}=Z^{(i),ss}_{\min}$ on
the blow-up. Using the notation of Definition 2.2, we introduce the following
schemes to define the centre of each blow-up.

######  Definition 5.16.

For a projective scheme $Y$ with an ample linearised $P$-action, we define

$C^{(i)}(Y):=C(Z^{(i),ss}_{\min}(Y),R^{(i)})\quad\text{and}\quad
B^{(i)}(Y):=p_{(i),Y}^{-1}(C^{(i)}(Y))\quad\quad\text{for }\>1\leq i\leq l-1.$

By definition, $C^{(i)}(Y)$ is the closed subscheme of $Z^{(i),ss}_{\min}(Y)$
on which the dimension of the $R^{(i)}$-stabiliser groups are maximal;
provided there is a strictly semistable point, this is a proper subscheme
disjoint from the stable locus $Z^{(i),s}_{\min}(Y)$. In the reductive partial
desingularisation procedure for the $R^{(i)}$-action on
$Z^{(i),ss}_{\min}(Y)$, this scheme (or its closure in $Z_{\min}^{(i)}(Y)$,
strictly speaking) is the centre of the first blow-up. Hence also $B^{(i)}(Y)$
is a closed subscheme of $Y_{\min}^{(i),ss}$.

######  Lemma 5.17.

The schemes $B^{(i)}(Y)$, and thus also their closures, are $P$-invariant.

###### Proof.

Let us just prove the case for $i=1$ as the other cases are similar. It
suffices to show invariance under the group $\widehat{H}=H\rtimes T$, as this
group surjects onto $P$. Since the central torus $T$ commutes with $R^{(1)}$,
we just need to show invariance under $H$. We argue as in Proposition 5.9: if
$h\in H$ and $y\in Y^{(1)}_{\min}$ then $p_{(1)}(hy)=h_{0}p_{(1)}(y)$ where
$h_{0}:=\lim_{t\rightarrow 0}\lambda^{(1)}(t)\cdot
h\cdot\lambda^{(1)}(t)^{-1}$. We can write this element as $h_{0}=h_{1}r_{1}$
where $r_{1}\in R^{(1)}$ and $h_{1}$ has zero first row except for an identity
matrix in the first block; thus $h_{1}$ commutes with $R^{(1)}$. Therefore,

$\operatorname{Stab}_{R^{(1)}}(p_{(1)}(hy))=\operatorname{Stab}_{R^{(1)}}(h_{1}r_{1}p_{(1)}(y))=r_{1}\operatorname{Stab}_{R^{(1)}}(p_{(1)}(y))r_{1}^{-1}$

from which the $H$-invariance follows. ∎

We can now define the first blow-up in this procedure, assuming that
$Z^{(1),s}_{\min}\neq Z^{(1),ss}_{\min}$.

######  Definition 5.18.

Let $\widehat{X}_{(1)}$ denote the blow-up of $X$ along the closure of
$B^{(1)}(X)$, with exceptional divisor denoted by $E_{(1)}$. As the centre of
this blow-up is $P$-invariant, there is an induced $P$-action on
$\widehat{X}_{(1)}$ which we linearise by the pulling back the linearisation
on $X$ and perturbing by a small multiple of the exceptional divisor $E^{1}$
as in [28].

######  Proposition 5.19.

Assume that $X$ satisfies (QiS) and (UU). If $Z^{(1),s}_{\min}\neq
Z^{(1),ss}_{\min}$, then the blow-up $\pi_{1}:\widehat{X}_{(1)}\rightarrow X$
has the following properties.

  1. _(a)_

$X^{s}_{\min}$ is disjoint from the centre of the blow-up
$\overline{B^{(1)}(X)}$,

  2. _(b)_

The strict transform of $Z_{\min}^{[i]}$ (resp. $Z_{\min}^{(i)}$) is
$Z_{\min}^{[i]}(\widehat{X}_{(1)})$ (resp.
$Z_{\min}^{(i)}(\widehat{X}_{(1)})$) for all $1\leq i\leq l-1$.

  3. _(c)_

$\widehat{X}_{(1)}$ also satisfies (QiS) and (UU) .

  4. _(d)_

The dimensions of $R^{(1)}$-stabilisers on the semistable
$\lambda^{(1)}$-minimal weight space drops:

$\max_{x\in{Z}_{\min}^{(1),ss}(\widehat{X}_{(1)})}\dim\operatorname{Stab}_{R^{(1)}}(x)<\max_{x\in
Z^{(1),ss}_{\min}}\dim\operatorname{Stab}_{R^{(1)}}(x).$

###### Proof.

For (a), we note that $Z^{(1),s}_{\min}$ is disjoint from $C^{(1)}(X)$, as
stable points have zero dimensional stabiliser and we assumed
$Z^{(1),s}_{\min}\neq Z^{(1),ss}_{\min}$. Thus
$B^{(1)}(X):=p_{(1)}^{-1}(C^{[1]}(X))$ is disjoint from $X^{(1),s}_{\min}$. We
then conclude (a) as $X^{s}_{\min}\subset X^{(1),s}_{\min}$ by Remark 4.35.
Note that $X^{s}_{\min}\neq\emptyset$, by Assumption (3) of (QiS).

For (b), we have seen that $Z^{s}_{\min}$ is disjoint from the centre of the
blow-up in (a) and since $Z^{s}_{\min}\subset Z^{[i],s}_{\min}$ (resp.
$Z^{s}_{\min}\subset Z^{(i),s}_{\min}$), we see that $Z^{[i]}_{\min}$ (resp.
$Z_{\min}$) is not contained in the centre of the blow-up. Consequently, we
deduce (b).

For (c), to deduce (UU) holds for $\widehat{X}_{(1)}$, we use Lemma 2.34 for
the groups $U^{[i]}$ and $U^{[i-1,i]}$, together with the fact that the
dimensions of these stabilisers on $X^{[i],ss}_{\min}$ are constant. We can
directly deduce that Assumptions (1) and (2) of (QiS) hold for
$\widehat{X}_{(1)}$ as the strict transform of $Z_{\min}^{[i]}$ (resp.
$Z_{\min}^{(i)}$) is $Z_{\min}^{[i]}(\widehat{X})$ (resp.
$Z_{\min}^{(i)}(\widehat{X})$) by (c). Finally, non-degeneracy (3) of (QiS)
holds for $\widehat{X}_{(1)}$ as we can identify $X^{s}_{\min}$ with a non-
empty open set of $\widehat{X}_{(1),\min}^{s}$ by (a).

For (d), we can use (b) to identify $Z_{\min}^{(1)}(\widehat{X})$ with the
strict transform of $Z_{\min}^{(1)}$. Then the result follows exactly as in
[28, Lemma 6.1 (iv)]. ∎

######  Proposition 5.20.

Assume that the $P$-action on $X$ satisfies (QiS) and (UU). Then there exists
a sequence of $P$-equivariant blow-ups resulting in a projective scheme
$\widehat{X}\rightarrow X$ such that $\widehat{X}$ satisfies (QiS) and (U).

###### Proof.

We iterate the above process for $R^{(1)}$: let $\widehat{X}_{(0)}=X$ and for
$i=1,\dots,n_{1}$, define $\widehat{X}_{(i)}$ to be the blow-up of
$\widehat{X}_{(i-1)}$ along the closure of $B^{(1)}(\widehat{X}_{(i-1)})$.
Since the dimensions of the $R^{(1)}$-stabilisers on the semistable locus in
the $\lambda^{(1)}$-minimal weight space decreases at each stage by
Proposition 5.19, this procedure terminates with a scheme
$\widehat{X}_{(n_{1})}$ for which stability and semistability coincides for
the $R^{(1)}$-action on the minimal $\lambda^{(1)}$-weight space.

We then turn our attention to $R^{(2)}$ and for $i=1,\dots,n_{2}$, define
$\widehat{X}_{(n_{1}+i)}$ to be the blow-up of $\widehat{X}_{(n_{1}+i-1)}$
along the closure of $B^{(2)}(\widehat{X}_{(n_{1}+i-1)})$. This terminates
with a scheme $\widehat{X}_{(n_{1}+n_{2})}$ for which stability and
semistability coincides for the $R^{(j)}$-action on the minimal
$\lambda^{(j)}$-weight space for $j\leq 2$. Once we have completed the blow-
ups for $R^{(j)}$, we turn our attention to $R^{(j+1)}$ and we progressively
construct a sequence of blow-ups

$\pi:\widehat{X}=\widehat{X}_{(n_{1}+\dots+n_{l-1})}\rightarrow\cdots\rightarrow\widehat{X}_{(n_{1}+n_{2})}\rightarrow\cdots\rightarrow\widehat{X}_{(n_{1})}\rightarrow\cdots\rightarrow
X$

such that stability and semistability coincides for the $R^{(j)}$-action on
the minimal $\lambda^{(j)}$-weight space in $\widehat{X}$ for $1\leq j\leq
l-1$; that is the Upstairs Reductive Stabiliser Assumption (UR) holds on
$\widehat{X}$. Furthermore, by Proposition 5.19 (c), (QiS) and (UU) also hold
for $\widehat{X}$. ∎

Now we complete the proof of Theorem 1.2 in the case where (UU) holds but (UR)
may fail.

######  Proposition 5.21.

Suppose that (QiS) and (UU) hold for the linearised $P$-action on $X$.

  1. (1)

If (UR) holds, we have $\operatorname{dom}(q)=X^{P-qs}$, which admits a
projective geometric $P$-quotient.

  2. (2)

If (UR) fails, there is an equivariant sequence of blow-ups
$\pi:\widehat{X}\rightarrow X$ such that $\widehat{X}$ satisfies the
conditions of (1). Moreover, $X^{P-qs}$ is isomorphic to an open subset of
$\widehat{X}^{P-qs}$ and thus $X^{P-qs}$ has a quasi-projective geometric
quotient by $\widehat{H}$, with a canonical projective completion given by the
geometric $P$-quotient of $\widehat{X}^{P-qs}$.

###### Proof.

In the case where (UR) holds, this is Proposition 5.14. If (UR) fails, we
perform the blow-up process described in Proposition 5.20 above, to obtain
$\pi:\widehat{X}\rightarrow X$ such that $\widehat{X}$ satisfies the
Quotienting-in-Stages and (U) (both reductive and unipotent). Hence,
$\widehat{X}$ satisfies the conditions of Proposition 5.14 and we obtain a
projective geometric $\widehat{H}$ quotient of the locus

$\widehat{X}^{P-qs}=\\{x\in\widehat{X}^{s}_{\min}\mid\widehat{p}_{[j]}(x)\notin
U^{[j]}\widehat{Z}_{\min}^{[j-1]}\text{ for all }1\leq j\leq l\\}.$

We have $\pi^{-1}(Z^{s}_{\min})\subset Z^{s}_{\min}(\widehat{X})$ and since by
Proposition 5.19 (a), $X^{s}_{\min}$ is disjoint from the centre of this blow-
up, we have also
$X^{s}_{\min}\cong\pi^{-1}(X^{s}_{\min})\subset\widehat{X}^{s}_{\min}$ . It
then follows that $X^{P-qs}\cong\pi^{-1}(X^{P-qs})\subset\widehat{X}^{P-qs}$,
and hence we get a quasi-projective geometric $\widehat{H}$-quotient of
$X^{P-qs}$ by restricting the quotient we obtained for $\widehat{X}^{P-qs}$. ∎

###  5.3. The case when the Upstairs Unipotent Stabiliser Assumption fails

We now conclude the proof of Theorem 1.2 section by proving Theorem 5.25,
which describes what to do when (UU) fails, but (WUU) holds.

Recall from Definition 4.11 that

$d^{[i]}_{\min}:=d_{\min}(X^{[i],ss}_{\min},U^{[i]})\quad\text{and}\quad
d^{[i-1,i]}_{\min}:=d_{\min}(X^{[i],ss}_{\min},U^{[i-1,i]})$

are respectively the minimal stabiliser dimensions of $U^{[i]}$ and
$U^{[i-1,i]}$ on $X^{[i],ss}_{\min}$. Similarly, we will use the notation
$d^{[i]}_{\max}$ and $d^{[i-1,i]}_{\max}$ for the maximal stabiliser
dimensions of these subgroups on $X^{[i],ss}_{\min}$. In the case already
considered, where (UU) holds, the dimensions of the stabilisers for $U^{[i]}$
and $U^{[i-1,i]}$ on $X^{[i],ss}_{\min}$ are constant, so these maximal and
minimal dimensions coincide. In the case where (UU) fails, we will use a blow-
up procedure to reduce to a situation where (UU) holds; however, to determine
the minimal weight spaces in the blow-ups we need to assume (WUU) holds.

Using the notation of Definition 2.2, we introduce the following schemes to
define the centre of each blow-up.

######  Definition 5.22.

For a projective scheme $Y$ with an ample linearised $P$-action, we define

$D^{[i]}(Y):=C(X^{[i],ss}_{\min}(Y),\widehat{U}^{[i]})\quad\text{and}\quad
D^{[i-1,i]}(Y):=C(X^{[i],ss}_{\min}(Y),\widehat{U}^{[i-1,i]})\quad\text{for
}\>1\leq i\leq l-1$

where
$\widehat{U}^{[i-1,i]}:=U^{[i-1,i]}\rtimes\lambda^{[i]}(\mathbb{G}_{m})$.

By definition, $D^{[i]}(Y)$ is the closed subscheme of $X^{[i],ss}_{\min}(Y)$
on which the dimension of the stabiliser for $\widehat{U}^{[i]}$ are maximal;
equivalently this is the $U^{[i]}$-sweep of $C(Z^{[i],ss}_{\min}(Y),U^{[i]})$.

######  Lemma 5.23.

The schemes $D^{[i]}(Y)$ and $D^{[i-1,i]}(Y)$, and thus also their closures,
are $P$-invariant.

###### Proof.

This follows as $U^{[i]}$ is a normal subgroup of $P$. Indeed, $U^{[i]}$ is
the unipotent radical of $P^{[i]}:=P(\lambda^{[i]})$ and so
$U^{[i]}\vartriangleleft P^{[i]}$. Since $P<P^{[i]}$, we deduce the claim. ∎

Let us examine the first step in the blow-up procedure. The assumption
[$Z_{\min}\not\subseteq C$] in (WUU) ensures that not all of $Z_{\min}$ is
blown up, allowing us to identify the minimal weight spaces in the blown up
space as the proper transforms of their counterparts on $X$.

######  Proposition 5.24.

Assume that $X$ satisfies (QiS) and (WUU). Suppose that
$d_{\min}^{[1]}<d_{\max}^{[1]}$. Then the blow-up
$\pi_{1}:\widetilde{X}_{(1)}\rightarrow X$ of $X$ along the closure $\mkern
1.5mu\overline{\mkern-1.5muD^{[1]}(X)\mkern-1.5mu}\mkern 1.5mu\subset X$ has
the following properties.

  1. _(a)_

The locus $C(X^{s}_{\min},U^{[1]},d^{[1]}_{\min})$ is non-empty and disjoint
from the centre of the blow-up. In particular, so is $X^{P-qs}$.

  2. _(b)_

The strict transform of $Z^{[1]}_{\min}$ in $\widetilde{X}$ coincides with the
blow-up $\widetilde{Z}_{(1)}$ of $Z^{[1]}_{\min}$ along the closure of
$C(Z^{[1],ss}_{\min},U^{[1]})$.

  3. _(c)_

The strict transform of $Z_{\min}^{[i]}$ (resp. $Z^{(i)}_{\min}$) is
$Z_{\min}^{[i]}(\widetilde{X}_{(1)})$ (resp.
$Z_{\min}^{(i)}(\widetilde{X}_{(1)})$) for $1\leq i\leq l-1$.

  4. _(d)_

The scheme $\widetilde{X}_{(1)}$ also satisfies (QiS) and (WUU).

  5. _(e)_

The dimensions of $U^{[1]}$-stabilisers in the $\lambda_{[1]}$-minimal weight
spaces drops:

$\max_{z\in{Z}_{\min}^{[1],ss}(\widetilde{X}_{(1)})}\dim\operatorname{Stab}_{U^{[1]}}(z)<\max_{z\in
Z^{[1],ss}_{\min}}\dim\operatorname{Stab}_{U^{[1]}}(z).$

###### Proof.

Assumption (WUU) implies that
$C(X^{s}_{\min},U^{[1]},d^{[1]}_{\min})\neq\emptyset$ and this is disjoint
from the centre of the blow-up, since only points with $U^{[1]}$-stabiliser
dimension equal to $d_{\max}^{[1]}$ are blown up. This proves (a).

To prove (b), first we claim that $Z^{[1]}_{\min}$ is not contained in the
centre of the blow-up. Recall that the centre of the blow up is

$D^{[1]}(X):=C(X^{[1],ss}_{\min}(X),\widehat{U}^{[1]})=U^{[1]}C(Z^{[1],ss}_{\min}(X),U^{[1]}).$

By Lemma 2.23 (see also [1] Lemma 5.2), we have for $u\in U^{[1]}$ and $z\in
Z^{[1]}_{\min}$ that $uz\in Z^{[1]}_{\min}$ if and only if $uz=z$. Hence, we
deduce that

$D^{[1]}(X)\cap Z^{[1]}_{\min}=C(Z^{[1],ss}_{\min}(X),U^{[1]}).$

Thus it suffices to show that $\mkern
1.5mu\overline{\mkern-1.5muC(Z^{[1],ss}_{\min}(X),U^{[1]})\mkern-1.5mu}\mkern
1.5mu\subsetneq Z^{[1]}_{\min}$ is a proper subscheme. But this consists only
of points with $U^{[1]}$-stabiliser dimension equal to $d_{\max}^{[1]}$, by
semicontinuity of stabiliser dimension. On the other hand, by (WUU) we know
that $C(Z_{\min},U^{[1]},d^{[1]}_{\min})\neq\emptyset$. Since $Z_{\min}\subset
Z_{\min}^{[1]}$ by (QiS) (1), we see that $Z_{\min}$ is not contained in the
centre of the blow-up and
$C(Z^{[1]}_{\min},U^{[1]},d^{[1]}_{\min})\neq\emptyset$. It then follows that
$Z^{[1]}_{\min}$ is not contained in the centre of the blow-up, which proves
(b).

To prove (c), the argument is the same as (b). By (WUU) we know that
$Z_{\min}$ is not contained in the centre of the blow-up and by (QiS) (1) we
have $Z_{\min}\subset Z^{[i]}_{\min}$ and $Z_{\min}\subset Z^{(i)}_{\min}$,
which shows both $Z^{[i]}_{\min}$ and $Z^{(i)}_{\min}$ are not contained in
the centre of the blow-up and proves (c).

For (d), as taking proper transforms preserves containment, Assumptions (1)
and (2) of (QiS) for $\widetilde{X}_{(1)}$ follow from the corresponding
properties for $X$ as (c) holds. For the non-degeneracy assumption (3) of
(QiS), take any $x\in X^{P-qs}$; this set is non-empty by (WUU). By (a),
$X^{P-qs}$ is disjoint from the centre of the blow-up, so we can use
$\pi_{1}^{-1}$ to lift $x\in X^{P-qs}$ to a unique
$\tilde{x}\in\widetilde{X}_{(1)}.$ We claim that
$\tilde{x}\in(\widetilde{X}_{(1)})^{s}_{\min}$. Let us use $\tilde{p}$,
$\tilde{p}_{[i]}$ to denote the corresponding maps for $\widetilde{X}_{(1)}$
rather than $X$. To prove our claim first note that, since $p(x)$ is not blown
up, we have $\tilde{p}(\tilde{x})\in Z_{\min}(\widetilde{X}_{(1)})$.
Furthermore, the linearisation on $\widehat{X}_{(1)}$ is an arbitrarily small
perturbation of the one on $X$, so $\tilde{p}(\tilde{x})$ is $R$-stable,
meaning that $\tilde{x}\in(\widetilde{X}_{(1)})^{s}_{\min}$. It follows that
$\tilde{x}$ is a point witnessing non-degeneracy for $\widetilde{X}_{(1)}$,
since otherwise $\pi_{1}(\tilde{x})$ could not lie in $X^{P-qs}$. Hence (QiS)
holds on $\widetilde{X}_{(1)}$.

For (WUU), observe that $\tilde{p}(\tilde{x})$ has the same stabilisers as its
image under $\pi_{1}$, since that image is not blown up. This shows that
[$Z_{\min}\not\subseteq C$] still holds. Moreover, the same is true for each
$\tilde{p}_{[i]}(\tilde{x})$ and its image under $\pi_{1}$, which together
with the above argument gives $\tilde{x}\in(\widetilde{X}_{(1)})^{P-qs}$, so
this set is non-empty. This proves (WUU) for $\widetilde{X}_{(1)}$, which
completes the proof of (d).

The proof of statement (e) is the same as that of Lemma 2.33. ∎

The above proposition gives the first step in the blow-up procedure for
$U^{[1]}$, which we can then iterate until $d^{[1]}_{\min}=d^{[1]}_{\min}$, at
which point we do the same for $U^{[2]}$, and so on until we reach
$U^{[l-1]}$. We then turn our attention to the groups $U^{[i-1,i]}$,
proceeding in the same way for those. At each stage in the blow-ups for the
groups $U^{[i]}$ and $U^{[i-1,i]}$, we need to ensure that not all of
$Z_{\min}$ is blown up so that the final blow-up satisfies (QiS): for this we
again use condition [$Z_{\min}\not\subseteq C$] of (WUU).

######  Theorem 5.25.

Assume that (QiS) and (WUU) hold for the linearised $P$-action on $X$. Then
the following statements hold.

  1. (1)

There exists a sequence of $P$-equivariant blow-ups resulting in a projective
scheme

$\pi:\widetilde{X}\rightarrow X$

such that $\widetilde{X}$ satisfies (QiS) and (UU).

  2. (2)

The Quotienting-in-Stages stable locus $X^{P-qs}$ admits a quasi-projective
geometric $P$-quotient with a natural projective completion.

###### Proof.

We inductively consider stabilisers for the groups $U^{[i]}$ and
$U^{[i-1,i]}$. The first group to consider is $U^{[1]}$ and the first blow-up
in this procedure is performed in Proposition 5.24. We iterate this procedure
for $U^{[1]}$: let $\widetilde{X}_{(0)}=X$ and for $i=1,\dots n_{1}$, define
$\widehat{X}_{(i)}$ to be the blow-up of $\widehat{X}_{(i-1)}$ along the
closure of $D^{[1]}(\widehat{X}_{(i-1)})$. Since the dimensions of the
$U^{[1]}$-stabilisers on the semistable locus in the $\lambda^{[1]}$-minimal
weight space decreases at each stage, this procedure terminates with a scheme
$\widehat{X}_{(n_{1})}$ which has constant dimensional $U^{[1]}$-stabilisers
on the semistable locus in the $\lambda^{[1]}$-minimal weight space equal to
$d^{[1]}_{\min}$. As in the proof of Proposition 2.35, it follows that
$\dim\operatorname{Stab}_{U^{[1]}}$ is constant and equal to $d^{[1]}_{\min}$
on all of $X^{[1],ss}_{\min}(\widehat{X}_{(n_{1})})$. Moreover, by repeated
application of Proposition 5.24, $\widehat{X}_{(n_{1})}$ satisfies (QiS) and
(WUU).

We then consider $U^{[2]}$ and for $i=1,\dots n_{2}$, define
$\widetilde{X}_{(n_{1}+i)}$ to be the blow-up of $\widetilde{X}_{(n_{1}+i-1)}$
along the closure of $D^{[2]}(\widetilde{X}_{(n_{1}+i-1)})$. By running the
proof of Proposition 5.24 with $U^{[1]}$ replaced by $U^{[2]}$, this
terminates with a scheme $\widetilde{X}_{(n_{1}+n_{2})}$ satisfying (QiS) and
(WUU), and such that for $i\leq 2$ we have $\dim\operatorname{Stab}_{U^{[i]}}$
is constant and equal to $d^{[i]}_{\min}$ on all of
$X^{[i],ss}_{\min}(\widehat{X}_{(n_{1}+n_{2})})$. Here, for $i=2$, this
follows from the blow-up construction, and for $i=1$, this is inherited from
$\widehat{X}_{(n_{1})}$ by Lemma 2.34.

After repeating this procedure for each $U^{[i]}$ we obtain a blow-up
$\widetilde{X}_{(N)}$ with $N=n_{1}+\cdots+n_{l-1}$ satisfying (QiS) and
(WUU), and such that for each $1\leq i\leq l-1$ we have

$\dim\operatorname{Stab}_{U^{[i]}}(x)=d^{[i]}_{\min}\text{ for all }x\in
X^{[i],ss}_{\min}(\widehat{X}_{(n_{1}+n_{2})});$

that is the stabiliser assumptions concerning each $U^{[i]}$ in (UU) are
satisfied.

We then consider the groups $U^{[i-1,i]}$ and inductively perform blow-ups
along the closures of $D^{[i-1,i]}(\widetilde{X}_{(N)})$ and so that the
dimensions of the $U^{[i-1,i]}$-stabilisers are constant on
$X^{[i],ss}_{\min}$. This procedure determines a sequence of blow-ups

$\pi:\widetilde{X}\rightarrow\cdots\rightarrow\widetilde{X}_{(1)}\rightarrow
X$

such that on $\widetilde{X}$ we have (QiS) and (UU), which proves (1).

For (2), since $X^{P-qs}$ is disjoint from the centre of each blow-up, we can
identify it with its preimage under $\pi$. In fact, by the argument in the
proof of Proposition 5.24 (d) we have
$\pi^{-1}(X^{P-qs})\subset\widetilde{X}^{P-qs}$. Now we can apply Proposition
5.21 to construct a quasi-projective geometric quotient of the open subset
$\widetilde{X}^{P-qs}$ of $\widetilde{X}$, possibly after performing a further
sequence of blow-ups $\widehat{X}\rightarrow\widetilde{X}$ by considering
reductive stabilisers as in §5.2.3. This restricts to the desired quasi-
projective geometric quotient of $X^{P-qs}$. The natural projective completion
is given by the projective $P$-quotient of $\widehat{X}$, which satisfies both
(QiS) and (U). ∎

We can now draw all of these results together into a proof of Theorem 1.2.

###### Proof of Theorem 1.2.

If both (QiS) and (U) hold, this is Proposition 5.14. If only (QiS) and (UU)
hold, this is Proposition 5.21. The final case, where only (QiS) and (WUU)
hold, is given by Theorem 5.25 above. ∎

###  5.4. Removing the irreducibility assumption

The irreducibility assumption for $X$ is mainly for bookkeeping purposes, and
is not necessary for our proof of Theorem 1.2 to work.

For reducible $X$, we first replace $X$ with the closure of $X_{\min}$, so
that $X_{\min}$ is still open and dense131313It may be in fact that the weight
space we are interested in is not the minimal one, in which case we can
replace $X$ with the closure of the corresponding attracting set. This may
happen, for example, when the non-degeneracy assumption 3 of (QiS) fails, or
when (UU) does not hold, _cf._ Remark 4.45. The proof of Theorem 2.29 is local
on affines, and so goes through verbatim, where we make sure to take the
affines small enough that they are irreducible. The rest of the argument, in
the case where no blow-ups are necessary, is the same.

The only other necessary modification to the proof is required considering
stabiliser dimensions for the reductive and non-reductive blow-up processes,
as there may be irreducible components of $X$ for which the generic stabiliser
dimension (for a particular group) is not equal to the global minimum
dimension. For this, as with the weight spaces, we can simply replace $X$ with
the union of those irreducible components having a point with global minimum
stabiliser dimension; once we have done this, the locus of points with minimum
stabiliser will be open and dense in the resulting space. Alternatively, as
discussed after Definition 2.2, one might prefer to think of the ‘minimum’
stabiliser dimension as being in fact a vector, recording the minimum on each
irreducible component.

##  6\. An overview of applications

The assumptions of Theorem 1.2 are suited to the construction of moduli spaces
of objects of fixed Harder-Narasimhan type in an abelian category (see Remark
4.30). In this section, we outline how to apply this theorem and discuss when
this can be applied to sheaves of fixed HN type on a polarised projective
scheme.

Given a moduli problem in an abelian category with a moduli space of
semistable objects constructed as a reductive GIT quotient of a parameter
scheme, to construct moduli spaces of objects of fixed Harder-Narasimhan type
$\tau$ one carries out the following procedure:

  1. (1)

Compare the HKKN stratification and HN stratification on the parameter scheme
appearing in the reductive GIT construction. Ideally identify the HN stratum
$S_{\tau}$ with (a closed subscheme of) a unstable HKKN stratum. This gives a
scheme $X_{\tau}$ (the closure of the subscheme $Y_{\tau}^{ss}\subset
S_{\tau}$ in the parameter space) with an action a parabolic subgroup
$P_{\tau}$, which we want to quotient by.

  2. (2)

Determine whether (QiS) holds: it should be straight-forward to verify the
first two conditions and the third, non-degeneracy, should admit a moduli-
theoretic interpretation similarly to Definition 4.38.

  3. (3)

Interpret the reductive stabiliser groups moduli-theoretically. For coprime HN
types (where semistability coincides with stability for all HN subquotients),
one may expect Assumption (UR) to hold similarly to the situation for sheaves.

  4. (4)

Interpret the unipotent stabiliser groups moduli theoretically and check
whether (UU), or the weaker assumption (WUU), holds.

  5. (5)

Give a moduli-theoretic interpretation of the quotienting-in-stages stable
locus $X_{\tau}^{P_{\tau}-qs}$ similarly to $\tau$-stability given in
Definition 4.18.

If all steps in this procedure can be achieved, one obtains a quasi-projective
moduli space of certain so-called $\tau$-stable objects of HN type $\tau$ and,
in good cases, this moduli space will even be projective.

Examples of moduli problems for which one could attempt to carry out the above
include moduli of sheaves, Higgs sheaves141414One difference to the standard
sheaves case is that in the Higgs case there are two different notions of HN
type: one can use either the HN type of the Higgs pair, or the HN type of the
underlying sheaf. Both will be considered in [18]. and representations of
quivers, which would require rephrasing as a projective GIT set-up in order to
apply results of NRGIT. In all these examples, the first of the above steps
have already been carried out in [15, 23, 21, 22, 17], and the second, third
and fifth are expected to be routine: the real crux is the fourth.

For example, fix a non-degenerate HN type $\tau$ of arbitrary length for
sheaves over a polarised projective base scheme $B$, and consider the problem
of constructing a moduli space for sheaves of HN type $\tau$. We have seen
above in $\S$4.4.2 and $\S$4.4.4 that, while Assumption (QiS) holds, the
Assumption (WUU) does not hold in general, which limits what we can presently
say. However, we can give a partial generalisation of the length two case in
[25] to higher lengths. First we need some definitions.

######  Definition 6.1.

(Refined HN types and refined non-degeneracy)

  1. (1)

A refined Harder-Narasimhan type $\widetilde{\tau}=(\tau,d)$ consists of a
Harder-Narasimhan type $\tau=(\tau_{1},\dots\tau_{l})$ and vector
$d=(d^{[1]},\dots d^{[l]},d^{[1,2]},\dots d^{[l-1,l]})$ of natural numbers. We
say that a sheaf $\mathcal{F}$ has refined HN type $\widetilde{\tau}$ if it
has HN type $\tau$ and we have

$\dim\operatorname{Hom}(\mathcal{F}/\mathcal{F}^{(i)},\mathcal{F}^{(i)})=d^{[i]}$
$\dim\operatorname{Hom}(\mathcal{F}/\mathcal{F}^{(i)},\mathcal{F}^{(i-1)})=d^{[j,j+1]}$

for all $i=1\dots l$ and $j=1\dots l-1$.

  2. (2)

By analogy with Definition 4.18, we say that a sheaf $\mathcal{F}$ of refined
HN type $\widetilde{\tau}$ is $\tilde{\tau}$-stable if each inclusion
$\mathcal{F}^{(i-1)}\subset\mathcal{F}^{(i)}$ in the HN filtration of
$\mathcal{F}$ is non-split and all subquotients
$\mathcal{F}^{(i)}/\mathcal{F}^{(i-1)}$ in the HN filtration are stable.

  3. (3)

By analogy with Definition 4.38 we say that $\widetilde{\tau}$ is non-
degenerate if there exists a $\widetilde{\tau}$-stable sheaf of refined HN
type $\widetilde{\tau}$.

######  Theorem 6.2.

Let $B$ be a polarised projective scheme and $\widetilde{\tau}=(\tau,d)$ be a
non-degenerate refined HN type for sheaves on $B$. If there exists a sheaf
$\mathcal{F}$ of type $\widetilde{\tau}$ such that
$\operatorname{gr}(\mathcal{F})\cong\mathcal{F}$, then there exists a quasi-
projective moduli space of $\tilde{\tau}$-stable sheaves.

###### Proof.

Consider the corresponding $P_{\tau}$-action on $X:=\overline{Y_{\tau}}$ as in
$\S$3.2.1. Let $Z_{\min}^{d}\subset Z_{\min}$ be the locus consisting of
points with stabiliser vector $d$, and let
$X^{d}_{\min}=p^{-1}(Z_{\min}^{d})\subset X$. Replacing $X$ with the closure
of $X^{d}_{\min}$, we obtain a projective scheme $W$ with a $P_{\tau}$-action,
for which we claim we can verify the conditions of Theorem 1.2. By non-
degeneracy of $\widetilde{\tau}$ we obtain non-degeneracy for $W$, and the
first two assumptions of (QiS) follow directly from Proposition 4.37. Using
Proposition 4.16 and Lemma 4.44 we deduce that $W^{P-qs}$ consists exactly of
the $\widetilde{\tau}$-stable sheaves. Hence (WUU) holds as
$W^{P-qs}\neq\emptyset$ and by assumption there exists a sheaf $\mathcal{F}$
of type $\widetilde{\tau}$ such that
$\operatorname{gr}(\mathcal{F})\cong\mathcal{F}$. Then the result follows by
applying Theorem 1.2. ∎

In particular, to obtain a moduli space for an refined HN type, we must assume
the existence of a graded sheaf of that type. A natural question is for which
extended HN types this assumption is satisfied, and in particular whether it
is satisfied for the minimum stabiliser vector for sheaves of a given HN type.

In the setting of Higgs sheaves of Higgs-HN type $\tau$ on a base scheme $S$,
one can show that, provided there exists a sheaf on $S$ of HN type $\tau$, it
is possible to construct a Higgs bundle $\mathcal{E}$ on $S$, such that
$\mathcal{E}\cong\operatorname{gr}^{\text{Higgs}-HN}(\mathcal{E})$ and this
has trivial unipotent stabiliser [18]. Therefore, via the spectral
correspondence (for example, see [35]), one can, at least in principle,
construct examples of base schemes $B$ from $S$ and HN types for sheaves from
the Higgs-HN type $\tau$ such that this assumption is satisfied for the
minimal stabiliser vector. The case of Higgs bundles will be dealt with in
more detail in upcoming work [18].

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