# A GIT construction of moduli spaces of sheaves of length 2

Yikun Qiao

###### Abstract

Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let
$\mathrm{Sch}/\Bbbk$ denote the category of schemes of finite type over
$\Bbbk$. Let $B$ be a connected projective scheme over $\Bbbk$ and let
$\mathcal{L}$ be an ample line bundle on $B$. Let $\tau$ be a Harder-
Narasimhan type of length 2, and let $\delta\in\mathbb{N}$. We say a pure
sheaf $\mathcal{E}$ on $B$ is _$(\tau,\delta)$ -stable_ if its Harder-
Narasimhan filtration $0=\mathcal{E}_{\leq 0}\subsetneq\mathcal{E}_{\leq
1}\subsetneq\mathcal{E}_{\leq 2}=\mathcal{E}$ is non-splitting, of type
$\tau$, with stable subquotients, and
$\delta=\dim_{\Bbbk}\mathrm{Hom}_{\mathcal{O}_{B}}(\mathcal{E}_{2},\mathcal{E}_{1})$
for $\mathcal{E}_{i}:=\mathcal{E}_{\leq i}/\mathcal{E}_{\leq i-1}$.

We define a moduli functor $\mathbf{M}^{\prime}_{\tau,\delta}$ classifying
$(\tau,\delta)$-stable sheaves on $B$ and construct its coarse moduli space by
non-reductive geometric invariant theory (GIT). We extend the non-reductive
GIT in [2] and [1] to linear actions on non-reduced schemes, and apply our
non-reductive GIT to prove that the sheafification
$(\mathbf{M}^{\prime}_{\tau,\delta})^{\sharp}$ on
$(\mathrm{Sch}/\Bbbk)_{\acute{e}tale}$ is represented by a quasi-projective
scheme.

Our methods generalise Jackson’s construction of moduli spaces of
$(\tau,\delta)$-stable sheaves in [11] in the category of varieties, to allow
non-reduced moduli schemes.

###### Contents

  1. 1 Introduction
  2. 2 The moduli problem and the GIT problem
    1. 2.1 The moduli problem
    2. 2.2 Simpson’s and Jackson’s moduli spaces
      1. 2.2.1 Moduli of stable and semistable sheaves
      2. 2.2.2 Moduli of unstable sheaves
  3. 3 Non-reductive GIT for graded additive groups
    1. 3.1 Some definitions for group actions
      1. 3.1.1 Infinitesimal stabilisers of scheme-valued points
      2. 3.1.2 Scheme-valued infinitesimal transformations
    2. 3.2 Some conditions for the linearisation
      1. 3.2.1 A non-emptiness condition
      2. 3.2.2 Upstairs Unipotent Stabiliser conditions
    3. 3.3 Theorem with Condition UU
      1. 3.3.1 Reduction to affine case
      2. 3.3.2 Affine theory with Condition UU
      3. 3.3.3 Gluing affine quotients
      4. 3.3.4 Quotients by $\hat{U}$ and parabolic groups
    4. 3.4 Theorem with Condition WUU
      1. 3.4.1 The blow-up centre
      2. 3.4.2 Main theorem
    5. 3.5 Stratification by unipotent stabilisers
  4. 4 Sheaves of length 2
    1. 4.1 Unipotent endomorphisms
    2. 4.2 The moduli functor $\mathbf{M}^{\prime}_{\tau,\delta}$
    3. 4.3 Local properties of the quotient
  5. A On the boundaries of $Y_{\tau}$ and $Y_{\tau}^{\mathrm{s}}$

##  1 Introduction

Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let
$\mathrm{Sch}/\Bbbk$ be the category of schemes of finite type over $\Bbbk$.
In this situation, a closed point of a scheme is equivalent to a
$\Bbbk$-point. Let $B$ be a connected projective scheme over $\Bbbk$ and let
$\mathcal{L}$ be an ample line bundle on $B$. Let $f:B\to\mathrm{Spec}(\Bbbk)$
be the structure morphism.

Let $P_{1}(t),P_{2}(t)\in\mathbb{Q}[t]$ be two numerical polynomials
satisfying

$\frac{P_{1}(t)}{\alpha_{1,d_{1}}}>\frac{P_{2}(t)}{\alpha_{2,d_{2}}}$ (1)

where $\alpha_{i,d_{i}}\in\mathbb{N}_{+}$ is the integer such that
$\frac{\alpha_{i,d_{i}}}{d_{i}!}t^{d_{i}}$ is the leading monomial of
$P_{i}(t)$, and the total order on $\mathbb{Q}[t]$ is the lexicographical
order. Let $\tau$ be a Harder-Narasimhan type of length 2 associated to
Hilbert polynomials $P_{1}(t),P_{2}(t)$, in the sense that we say a pure sheaf
$\mathcal{E}$ on $B$ has type $\tau$ if $\mathcal{E}_{i}:=\mathcal{E}_{\leq
i}/\mathcal{E}_{\leq i-1}$ is semistable of Hilbert polynomial $P_{i}(t)$ for
$i=1,2$, where

$0=\mathcal{E}_{\leq 0}\subsetneq\mathcal{E}_{\leq
1}\subsetneq\mathcal{E}_{\leq 2}=\mathcal{E}$ (2)

is the Harder-Narasimhan filtration of $\mathcal{E}$. Let
$\delta\in\mathbb{N}$, which we choose as another discrete invariant. For a
pure sheaf $\mathcal{E}$ of type $\tau$, we say that the space of _unipotent
endomorphisms_ of $\mathcal{E}$ has dimension $\delta$ if
$\delta=\dim_{\Bbbk}\mathrm{Hom}_{\mathcal{O}_{B}}(\mathcal{E}_{2},\mathcal{E}_{1})$.

Jackson has studied the moduli problem of _$(\tau,\delta)$ -stable sheaves_ in
[11]. A sheaf $\mathcal{E}$ on $B$ is called _$(\tau,\delta)$ -stable_ if

  * •

$\mathcal{E}$ has type $\tau$;

  * •

$\mathcal{E}_{i}$ is stable for $i=1,2$;

  * •

$\mathcal{E}\not\cong\mathcal{E}_{1}\oplus\mathcal{E}_{2}$;

  * •

$\delta=\dim\mathrm{Hom}_{\mathcal{O}_{B}}(\mathcal{E}_{2},\mathcal{E}_{1})$.

He used non-reductive geometric invariant theory developed in [2] and [1] to
construct moduli spaces of $(\tau,\delta)$-stable sheaves. However, the non-
reductive GIT in [2] and [1] is for actions on _reduced_ projective schemes,
while the desired moduli spaces may not be necessarily reduced (See [3] Remark
7.8. ). The paper aims to extend results in [11] to allow potentially non-
reduced moduli schemes.

For $T\in\mathrm{Sch}/\Bbbk$, we call a coherent sheaf $\mathcal{E}$ on
$B_{T}:=B\times_{\Bbbk}T$ a _flat family over $T$ of $(\tau,\delta)$-stable
sheaves on $B$_ if:

  * •

there is a length 2 filtration of $\mathcal{O}_{B_{T}}$-submodules

$0=\mathcal{E}_{\leq 0}\subsetneq\mathcal{E}_{\leq
1}\subsetneq\mathcal{E}_{\leq 2}=\mathcal{E}$ (3)

such that $\mathcal{E}_{i}:=\mathcal{E}_{\leq i}/\mathcal{E}_{\leq i-1}$ is a
flat family over $T$ of stable sheaves of Hilbert polynomial $P_{i}(t)$ for
$i=1,2$;

  * •

$\mathcal{E}_{\leq 1}\subsetneq\mathcal{E}_{\leq 2}$ does not split at any
closed $t\in T$;

  * •

for every closed subscheme $T^{\prime}\hookrightarrow T$, the following
$\mathcal{O}_{T^{\prime}}$-module is locally free of rank $\delta$

$(B_{T^{\prime}}\to
T^{\prime})_{*}\mathcal{H}om_{\mathcal{O}_{B_{T^{\prime}}}}\big{(}\mathcal{E}|_{B_{T^{\prime}}},\mathcal{E}_{1}|_{B_{T^{\prime}}}\big{)}.$
(4)

We define a moduli functor
$\mathbf{M}^{\prime}_{\tau,\delta}:(\mathrm{Sch}/\Bbbk)^{\mathrm{op}}\to\mathrm{Set}$
which to $T\in\mathrm{Sch}/\Bbbk$ assigns the set

$\mathbf{M}^{\prime}_{\tau,\delta}(T):=\Big{\\{}\begin{matrix}\textrm{isomorphism
classes of flat families}\\\ \textrm{over }T\textrm{ of
}(\tau,\delta)\textrm{-stable sheaves on }B\end{matrix}\Big{\\}}.$ (5)

The final result of this paper is the following.

######  Theorem.

Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let
$\mathrm{Sch}/\Bbbk$ denote the category of schemes of finite type over
$\Bbbk$. Let $\tau$ be a Harder-Narasimhan type of length 2. Let
$\delta\in\mathbb{N}$. Then there exists a quasi-projective scheme
$M_{\tau,\delta}\in\mathrm{Sch}/\Bbbk$ representing
$(\mathbf{M}^{\prime}_{\tau,\delta})^{\sharp}$, the sheafification of
$\mathbf{M}^{\prime}_{\tau,\delta}$ with respect to the étale topology. In
particular, the moduli functor $\mathbf{M}^{\prime}_{\tau,\delta}$ has a
coarse moduli space $M_{\tau,\delta}$.

In Section 2, we first introduce the notion of _flat families of
$(\tau,\delta)$-stable sheaves on $B$_ parametrised by schemes (Definition
2.3), which generalises the notion of $(\tau,\delta)$-stable sheaves. We
consider the moduli functor $\mathbf{M}^{\prime}_{\tau,\delta}$ sending
$T\in\mathrm{Sch}/\Bbbk$ to the set of isomorphism classes of flat families
over $T$ of $(\tau,\delta)$-stable sheaves on $B$. Next, we set up the non-
reductive GIT problem in the category of schemes. For $m\gg 0$ and
$V:=\Bbbk^{P(m)}$, there exist locally closed subschemes
$Y_{\tau}^{\mathrm{(s)s}}\subseteq
Y_{\tau}\subseteq\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$, where
$V(-m):=V\otimes_{\Bbbk}\mathcal{L}^{-m}$. The schemes
$Y_{\tau}^{\mathrm{(s)s}}\subseteq Y_{\tau}$ are determined by their functors
of points. For example, a morphism $T\to Y_{\tau}^{\mathrm{s}}$ is equivalent
to an isomorphism class of a flat family over $T$ of quotients of
$V(-m):=V\otimes_{\Bbbk}\mathcal{L}^{-m}$

$q:V(-m)_{T}\to\mathcal{E},\quad V(-m)_{T}:=(B_{T}\to
B)^{*}\big{(}V(-m)\big{)}$ (6)

denoted by $[(\mathcal{E},q)]$, such that $\mathcal{E}$ is a flat family over
$T$ of sheaves of type $\tau$ with stable subquotients. Let
$\overline{Y_{\tau}^{\mathrm{s}}}\subseteq\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$
denote the scheme theoretic closure. Let
$\mathrm{P}_{\tau}\subseteq\mathrm{SL}(V)$ be the parabolic subgroup
stabilising the partial flat $0=V_{\leq 0}\subseteq V_{\leq 1}\subseteq
V_{\leq 2}=V$, where $V_{i}:=\Bbbk^{P_{i}(m)}$ and $V_{\leq
i}:=V_{1}\oplus\cdots\oplus V_{i}$. Recall that
$\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$ has a very ample line
bundle $\mathcal{M}$. The non-reductive GIT problem we consider is

$\mathrm{P}_{\tau}\curvearrowright\big{(}\overline{Y_{\tau}^{\mathrm{s}}},\mathcal{M}\big{|}_{\overline{Y_{\tau}^{\mathrm{s}}}}\big{)}.$
(7)

This is an enhancement of the non-reductive GIT problem considered in [11],
which was essentially the reduction of the linearisation above.

In Section 3, we develop the non-reductive GIT for groups of the form
$\hat{U}:=U\rtimes\mathbb{G}_{m}$, where $U:=(\mathbb{G}_{a})^{r}$ is additive
and $\mathbb{G}_{m}\to\mathrm{Aut}(U)$ induces a representation of
$\mathbb{G}_{m}$ on $\mathfrak{u}:=\mathrm{Lie}(U)$ with a unique weight which
is positive. Non-reductive GIT for linear $\hat{U}$-actions on reduced
projective schemes has been considered in [2], [1] and [6]. We follow their
ideas and generalise to linear $\hat{U}$-actions on non-reduced projective
schemes. Let $X$ be a projective scheme and let $L$ be an ample line bundle on
$X$. Let $\hat{U}$ act linearly on $X$ with respect to $L$.

There is a morphism
$\phi:\Omega_{X/\Bbbk}\to\mathfrak{u}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}$ and
we call $\phi$ the _infinitesimal action of $\mathfrak{u}$ on $X$_. There are
important Condition UU and Condition WUU, generalising [6] Assumption 4.40 and
4.42 for $\hat{U}\curvearrowright(X,L)$. Condition UU assumes that
$X^{0}_{\min}\neq\emptyset$ and that $\mathrm{coker}(\phi)|_{X^{0}_{\min}}$ is
locally free of constant rank. Condition WUU assumes roughly that
$\mathrm{coker}(\phi)$ is locally free of the lowest possible rank on an open
subset which intersects with $Z_{\min}:=(X^{0}_{\min})^{\mathbb{G}_{m}}$,
where $Z_{\min}$ is defined as the _fixed point subscheme_ in the sense of
[4]. These conditions reduce to [6] Assumption 4.40 and 4.42 when $X$ is
reduced.

When Condition UU holds, the open subscheme $X^{0}_{\min}$ has a quasi-
projective universal geometric quotient by $U$ (Theorem 3.24), and
$X^{0}_{\min}\setminus UZ_{\min}$ has a projective universal geometric
quotient by $\hat{U}$ (Theorem 3.27). Moreover the quotient $X^{0}_{\min}/U$
represents the associated quotient sheaf on
$(\mathrm{Sch}/\Bbbk)_{\acute{e}tale}$ (Corollary 3.26). When Condition WUU
holds, we construct a $\hat{U}$-equivariant blowing up $b:\widetilde{X}\to X$
such that Condition UU holds for
$\hat{U}\curvearrowright\big{(}\widetilde{X},\widetilde{L}\big{)}$, where
$\widetilde{L}:=\mathcal{O}_{\widetilde{X}}(-E)\otimes_{\mathcal{O}_{\widetilde{X}}}b^{*}L^{a}$
for $a\gg 0$ and $E\subseteq\widetilde{X}$ is the exceptional divisor.

In general, we only assume $X^{0}_{\min}\neq\emptyset$. We consider the
_stratification by unipotent stabilisers_
$\coprod_{\delta\in\mathbb{N}}S_{\delta}(X^{0}_{\min})\to X^{0}_{\min}$, where
$S_{\delta}(X^{0}_{\min})\subseteq X^{0}_{\min}$ is the locally closed
subscheme which is universal to the property that
$\mathrm{coker}(\phi)|_{S_{\delta}(X^{0}_{\min})}$ is locally free of rank
$\delta$. If $S_{\delta}(X^{0}_{\min})\cap Z_{\min}\neq\emptyset$, then
Condition WUU holds for
$\hat{U}\curvearrowright\big{(}\overline{S_{\delta}(X^{0}_{\min})},L\big{|}_{\overline{S_{\delta}(X^{0}_{\min})}}\big{)}$,
where $\overline{S_{\delta}(X^{0}_{\min})}\subseteq X$ is the scheme theoretic
closure. We can apply our results with Condition WUU to get, for example,
Theorem 3.33 and Theorem 3.34.

In Section 4, we apply our non-reductive GIT for the moduli problem
$\mathbf{M}^{\prime}_{\tau,\delta}$. We have defined
$Y_{\tau}^{\mathrm{(s)s}}\subseteq Y_{\tau}$ in Section 2. Let
$Y_{\tau,\delta}^{\mathrm{s}}:=S_{\delta}(Y_{\tau}^{\mathrm{s}})\setminus\mathrm{U}_{\tau}Z_{\min}(\overline{Y_{\tau}^{\mathrm{s}}})$,
which is open in $S_{\delta}(Y_{\tau}^{\mathrm{s}})$. Similar to
$Y_{\tau}^{\mathrm{(s)s}}\subseteq Y_{\tau}$, we can descibe
$Y_{\tau,\delta}^{\mathrm{s}}\subseteq S_{\delta}(Y_{\tau}^{\mathrm{s}})$ by
their functors of points. For a morphism
$T\to\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$, it factors through
$Y_{\tau,\delta}^{\mathrm{s}}$ if the associated class of families
$[(\mathcal{E},q)]$ over $T$ satisfies that $\mathcal{E}$ is a flat family
over $T$ of $(\tau,\delta)$-stable sheaves (Lemma 4.2). This links the moduli
problem $\mathbf{M}^{\prime}_{\tau,\delta}$ with the GIT problem in Section 2.
By Theorem 3.34, the action $\mathrm{P}_{\tau}\curvearrowright
Y_{\tau,\delta}^{\mathrm{s}}$ has a quasi-projective universal geometric
quotient $Y_{\tau,\delta}^{\mathrm{s}}/\mathrm{P}_{\tau}$. The quotient
$Y_{\tau,\delta}^{\mathrm{s}}\to
Y_{\tau,\delta}^{\mathrm{s}}/\mathrm{P}_{\tau}$ is the composition of a
$\mathrm{U}_{\tau}$-quotient and a
$\mathrm{P}_{\tau}/\mathrm{U}_{\tau}$-quotient. In both stages, the GIT
quotients represent the associated quotient sheaves on
$(\mathrm{Sch}/\Bbbk)_{\acute{e}tale}$. This observation together with a
standard categorical argument (for example [9] Lemma 4.3.1) shows that
$(\mathbf{M}^{\prime}_{\tau,\delta})^{\sharp}\cong
Y_{\tau,\delta}^{\mathrm{s}}/\mathrm{P}_{\tau}$.

##  2 The moduli problem and the GIT problem

Recall that $\mathrm{Sch}/\Bbbk$ is the category of schemes of finite type
over $\Bbbk$, and $B$ is a projective scheme over $\Bbbk$ with an ample line
bundle $\mathcal{L}$. Recall $P_{i}(t),P(t)\in\mathbb{Q}[t]$ are prescribed
Hilbert polynomials.

For an integer $n\in\mathbb{Z}$ and a vector space $V$, let
$V(-n):=V\otimes_{\Bbbk}\mathcal{L}^{-n}$ denote the coherent sheaf on $B$
twisted from the constant sheaf by $\mathcal{L}^{-n}$. For
$T\in\mathrm{Sch}/\Bbbk$ and $\mathcal{F}$ a coherent sheaf on $B$, let
$B_{T}:=B\times_{\Bbbk}T$ denote the base change of $B$ and let
$\mathcal{F}_{T}:=(B_{T}\to B)^{*}\mathcal{F}$ denote the base change of
$\mathcal{F}$. Consider $\mathrm{Quot}_{V(-n)/B/\Bbbk}^{P,\mathcal{L}}$, the
Quot scheme parametrising quotients $q:V(-n)\to\mathcal{E}$ such that the
$\mathcal{E}$ has Hilbert polynomial $P(t)\in\mathbb{Q}[t]$ with respect to
$\mathcal{L}$.

###  2.1 The moduli problem

Let $T\in\mathrm{Sch}/\Bbbk$. Recall that $\tau=(P_{1},P_{2})$ is a Harder-
Narasimhan type, and we say a coherent sheaf $\mathcal{E}$ on $B$ has type
$\tau$ if and only if $\mathcal{E}$ is pure and there exists a length 2
filtration

$0=\mathcal{E}_{\leq 0}\subsetneq\mathcal{E}_{\leq
1}\subsetneq\mathcal{E}_{\leq 2}=\mathcal{E}$ (8)

such that $\mathcal{E}_{i}:=\mathcal{E}_{\leq i}/\mathcal{E}_{\leq i-1}$ is
semistable of Hilbert polynomial $P_{i}(t)$ for $i=1,2$.

We extend the notion of type $\tau$ Harder-Narasimhan filtration for flat
families over $T\in\mathrm{Sch}/\Bbbk$.

######  Definition 2.1.

For $T\in\mathrm{Sch}/\Bbbk$, let $\mathcal{E}$ be a coherent
$\mathcal{O}_{B_{T}}$-module flat over $T$. A _type $\tau$ Harder-Narasimhan
filtration_ of $\mathcal{E}$ is a filtration of coherent
$\mathcal{O}_{B_{T}}$-modules

$0=\mathcal{E}_{\leq 0}\subsetneq\mathcal{E}_{\leq
1}\subsetneq\mathcal{E}_{\leq 2}=\mathcal{E}$ (9)

such that $\mathcal{E}_{i}:=\mathcal{E}_{\leq i}/\mathcal{E}_{\leq i-1}$ is a
flat family over $T$ of semistable sheaves of Hilbert polynomial $P_{i}(t)$
for $i=1,2$.

We say $\mathcal{E}$ is a _flat family over $T$ of type $\tau$ sheaves on $B$_
if it admits a type $\tau$ Harder-Narasimhan filtration.

######  Lemma 2.2.

For $T\in\mathrm{Sch}/\Bbbk$ and a coherent $\mathcal{O}_{B_{T}}$-module
$\mathcal{E}$. If a type $\tau$ Harder-Narasimhan filtration exits, then the
filtration is unique.

For a morphism $f:S\to T$ in $\mathrm{Sch}/\Bbbk$ and a coherent
$\mathcal{O}_{B_{T}}$-module $\mathcal{E}$ with type $\tau$ Harder-Narasimhan
filtration $\\{\mathcal{E}_{\leq i}\\}_{i=0}^{2}$, the pullback $(1_{B}\times
f)^{*}\mathcal{E}$ has a type $\tau$ Harder-Narasimhan filtration, which is
the pullback of $\\{\mathcal{E}_{\leq i}\\}_{i=0}^{2}$.

###### Proof.

This is a special case of [13] Theorem 5, where the filtration
$\\{\mathcal{E}_{\leq i}\\}_{i=0}^{2}$ is called the _relative_ Harder-
Narasimhan filtration of $\mathcal{E}$. ∎

######  Definition 2.3.

For $\delta\in\mathbb{N}$ and $T\in\mathrm{Sch}/\Bbbk$, a _flat family over
$T$ of $(\tau,\delta)$-stable sheaves on $B$_ is a flat family over $T$ of
type $\tau$ sheaves $\mathcal{E}$, which has a Harder-Narasimhan filtration

$0=\mathcal{E}_{\leq 0}\subsetneq\mathcal{E}_{\leq
1}\subsetneq\mathcal{E}_{\leq 2}=\mathcal{E}$ (10)

satisfying:

  * •

$\mathcal{E}_{i}=\mathcal{E}_{\leq i}/\mathcal{E}_{\leq i-1}$ is a flat family
over $T$ of stable sheaves of Hilbert polynomial $P_{i}(t)\in\mathbb{Q}[t]$,
for $i=1,2$;

  * •

$\mathcal{E}_{\leq 1}\subseteq\mathcal{E}_{\leq 2}$ does not split at any
closed point $t\in T$;

  * •

for every closed subscheme $T^{\prime}\subseteq T$, the quasi-coherent
$\mathcal{O}_{T^{\prime}}$-module

$(B_{T^{\prime}}\to
T^{\prime})_{*}\mathcal{H}om_{\mathcal{O}_{B_{T^{\prime}}}}\big{(}\mathcal{E}_{2}|_{B_{T^{\prime}}},\mathcal{E}_{1}|_{B_{T^{\prime}}}\big{)}$
(11)

is locally free of rank $\delta$.

######  Remark 2.4.

Let $T=\mathrm{Spec}(\Bbbk)$ and let $d\in\mathbb{N}$. A flat family over $T$
of type $\tau$ sheaves is exactly a pure $\mathcal{O}_{B}$-module
$\mathcal{E}$ of Harder-Narasimhan type $\tau$. A flat family over $T$ of
$(\tau,\delta)$-stable sheaves is exact a pure $\mathcal{O}_{B}$-module
$\mathcal{E}$ of type $\tau$ such that $\mathcal{E}_{1},\mathcal{E}_{2}$ are
stable and $\mathcal{E}\not\cong\mathcal{E}_{1}\oplus\mathcal{E}_{2}$ and
$\mathrm{Hom}_{\mathcal{O}_{B}}(\mathcal{E}_{2},\mathcal{E}_{1})$ has
dimension $\delta$.

The moduli functor $\mathbf{M}^{\prime}_{\tau,\delta}$ is defined as follows

$\mathbf{M}^{\prime}_{\tau,\delta}:(\mathrm{Sch}/\Bbbk)^{\mathrm{op}}\to\mathrm{Set}$
(12)

which sends $T\in\mathrm{Sch}/\Bbbk$ to the set

$\mathbf{M}^{\prime}_{\tau,\delta}(T):=\Big{\\{}\begin{matrix}\textrm{isomorphism
classes of flat families}\\\ \textrm{over }T\textrm{ of
}(\tau,\delta)\textrm{-stable sheaves on }B\end{matrix}\Big{\\}}$ (13)

and for a morphism $g:T^{\prime}\to T$, the map

$\mathbf{M}^{\prime}_{\tau,\delta}(g):\mathbf{M}^{\prime}_{\tau,\delta}(T)\to\mathbf{M}^{\prime}_{\tau,\delta}(T^{\prime})$
(14)

sends the isomorphism class of $\mathcal{E}$ to the isomorphism class of the
class of $\mathcal{E}|_{B_{T^{\prime}}}$, where
$\mathcal{E}|_{\mathcal{B}_{T^{\prime}}}:=(1_{B}\times g)^{*}\mathcal{E}$. Let
$\mathrm{P}_{\tau}$ act trivially on $\mathbf{M}^{\prime}_{\tau,\delta}$.

Consider the étale site $(\mathrm{Sch}/\Bbbk)_{\acute{e}tale}$. We will show
that the sheafification $(\mathbf{M}^{\prime}_{\tau,\delta})^{\sharp}$ is
representable (Theorem 4.6). The representative is the coarse moduli space of
$\mathbf{M}^{\prime}_{\tau,\delta}$. The moduli space can be constructed by
non-reductive GIT, which is quasi-projective.

###  2.2 Simpson’s and Jackson’s moduli spaces

We introduce Simpson’s moduli spaces of stable and semistable sheaves ([14])
and Jackson’s moduli spaces of sheaves of Harder-Narasimhan length 2 ([11]).
These moduli spaces were constructed by reductive GIT and non-reductive GIT
respectively. These moduli spaces were coarse moduli spaces of naturally
defined moduli functors, so they are independent of the way we set up the GIT
problems. Moduli spaces of stable sheaves have natural completions, the
corresponding moduli spaces of semistable sheaves. In [11], completions of the
moduli spaces of sheaves of Harder-Narasimhan length 2 were described as non-
reductive quotients of blow-ups of the associated GIT problem. However, it is
unclear now whether the completion is independent of the GIT problem, and
there is no natural moduli functor associated to the completion yet.

####  2.2.1 Moduli of stable and semistable sheaves

Let $B$ be a connected projective scheme over $\Bbbk$ with an ample line
bundle $\mathcal{L}$. Let $P(t)\in\mathbb{Q}[t]$ be a numerical polynomial.
For $j\in\mathbb{Z}$ and a coherent sheaf $\mathcal{F}$ on $B$, we denote
$\mathcal{F}(j):=\mathcal{F}\otimes_{\mathcal{O}_{B}}\mathcal{L}^{j}$. Moduli
spaces of semistable pure sheaves on $B$ of fixed Hilbert polynomials were
constructed by Simpson in [14]. A more recent reference is [9].

Let $\mathrm{Sch}/\Bbbk$ be the category of schemes of finite type over
$\Bbbk$. For $T\in\mathrm{Sch}/\Bbbk$, a _family over $T$ of (semi)stable
sheaves of Hilbert polynomial $P(t)$_ is a coherent sheaf $\mathcal{F}$ on
$B_{T}:=B\times_{\Bbbk}T$ such that $\mathcal{F}$ is flat over $T$ and
$\mathcal{F}_{s}$ is a (semi)stable sheaf on $B$ of Hilbert polynomial $P(t)$
for every closed point $s\in T$, where $\mathcal{F}_{s}$ is the pullback of
$\mathcal{F}$ along $B\cong
B\times_{\Bbbk}\mathrm{Spec}(\kappa(s))\hookrightarrow B_{T}$. Two families
$\mathcal{F},\mathcal{F}^{\prime}$ on $T$ are called equivalent if there
exists a line bundle $\mathcal{A}$ on $T$ such that
$\mathcal{F}\cong\mathcal{F}^{\prime}\otimes_{\mathcal{O}_{B_{T}}}(f_{T})^{*}\mathcal{A}$,
where $f_{T}:B_{T}\to T$ is the pullback of $f:B\to\mathrm{Spec}(\Bbbk)$. With
the notions of families and their equivalence relations, we consider the
moduli functors

$\begin{split}\mathbf{M}_{B/\Bbbk}^{P(t),\mathcal{L},\mathrm{ss}}:&(\mathrm{Sch}/\Bbbk)^{\mathrm{op}}\to\mathrm{Set}\\\
\mathbf{M}_{B/\Bbbk}^{P(t),\mathcal{L},\mathrm{s}}:&(\mathrm{Sch}/\Bbbk)^{\mathrm{op}}\to\mathrm{Set}\end{split}$
(15)

which associate to $T\in\mathrm{Sch}/\Bbbk$ the sets

$\begin{split}\mathbf{M}_{B/\Bbbk}^{P(t),\mathcal{L},\mathrm{(s)s}}(T):=&\bigg{\\{}\begin{matrix}\textrm{families
on }T\textrm{ of (semi)stable sheaves}\\\ \textrm{of Hilbert polynomial
}P(t)\end{matrix}\bigg{\\}}\bigg{/}\textrm{equivalence relation}\end{split}$
(16)

and the functors on morphisms are naturally pullbacks of coherent sheaves.
These functors are both corepresentable and
$\mathbf{M}_{B/\Bbbk}^{P(t),\mathcal{L},\mathrm{s}}$ has a coarse moduli
space.

By [14] Theorem 1.1 and [9] Theorem 3.3.7, the collection of semistable
sheaves with Hilbert polynomial $P(t)$ is bounded. By [9] Lemma 1.7.6, their
regularities are uniformly bounded above. For $m\gg 0$, all semistable sheaves
$\mathcal{F}$ of Hilbert polynomial $P(t)$ satisfy:

  * •

$\dim H^{0}(B,\mathcal{F}(m))=P(m)$;

  * •

$H^{0}(B,\mathcal{F}(m))\otimes_{\Bbbk}\mathcal{L}^{-m}\to\mathcal{F}$ is
surjective.

Let $V=\Bbbk^{P(m)}$. There is a non-canonical linear isomorphism $V\cong
H^{0}(B,\mathcal{F}(m))$. With one such isomorphism, we have a surjective
morphism

$q:V(-m)\to\mathcal{F}$ (17)

which represents a $\Bbbk$-point in
$\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$.

Let $R\subseteq\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$ be the open
subscheme whose closed points are classes $[(\mathcal{F},q)]$ such that:

  * (1)

$\mathcal{F}$ is semistable;

  * (2)

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

Let $R^{\mathrm{s}}\subseteq R$ be the open subset of classes
$[(\mathcal{F},q)]$ such that $\mathcal{F}$ is stable. Condition $(2)$ above
is open since the function $[(\mathcal{F},q)]\mapsto\dim H^{0}(B,\ker(q)(m))$
is upper semi-continuous and the condition that $\dim H^{0}(B,\ker(q)(m))=0$
is equivalent to the condition $(2)$. Condition $(1)$ above is open by [9]
Proposition 2.3.1. This justifies the claim that
$R\subseteq\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$ is open, and
similarly $R^{\mathrm{s}}\subseteq R$ is open.

Recall that for $M\gg m\gg 0$, there is a closed immersion

$\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}\hookrightarrow\mathrm{Grass}\big{(}V\otimes_{\Bbbk}H,P(M)\big{)}$
(18)

where $H:=H^{0}(B,\mathcal{L}^{M-m})$. This is the classifying morphism of the
quotient on $\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$

$V\otimes_{\Bbbk}H^{0}(B,\mathcal{L}^{M-m})\otimes_{\Bbbk}\mathcal{O}_{\mathrm{Quot}}=f_{\mathrm{Quot},*}\big{(}V_{\mathrm{Quot}}(M-m)\big{)}\to
f_{\mathrm{Quot},*}\mathcal{U}(M)$ (19)

where $V(-m)_{\mathrm{Quot}}\to\mathcal{U}$ is the universal quotient on
$B_{\mathrm{Quot}}=B\times_{\Bbbk}\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$,
and
$f_{\mathrm{Quot}}:B_{\mathrm{Quot}}\to\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$.

The Plücker immersion is a closed immersion

$\mathrm{Grass}\big{(}V\otimes_{\Bbbk}H,P(M)\big{)}\hookrightarrow\mathbb{P}^{N}:=\mathbb{P}\bigg{(}\Big{(}\bigwedge^{P(M)}\big{(}V\otimes_{\Bbbk}H\big{)}\Big{)}^{*}\bigg{)}.$
(20)

The very ample line bundle $\mathcal{O}_{\mathbb{P}^{N}}(1)$ pulls back to a
very ample line bundle on $\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$,
denoted by $\mathcal{M}$. Then we have

$\mathcal{M}=\det\bigg{(}f_{\mathrm{Quot},*}\Big{(}\mathcal{U}\otimes_{\mathcal{O}_{B_{\mathrm{Quot}}}}\mathcal{L}_{\mathrm{Quot}}^{\otimes
M}\Big{)}\bigg{)}.$ (21)

There is an action of $\mathrm{GL}(V)$ on
$\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$, which is determined by
maps

$h_{\mathrm{GL}(V)}(T)\times h_{\mathrm{Quot}}(T)\to
h_{\mathrm{Quot}}(T),\quad T\in\mathrm{Sch}/\Bbbk.$ (22)

Let $(g,q)\in h_{\mathrm{GL}(V)}(T)\times h_{\mathrm{Quot}}(T)$. The morphism
$g:T\to\mathrm{GL}(V)$ corresponds to an invertible morphism
$g:V\otimes_{\Bbbk}\mathcal{O}_{T}\to V\otimes_{\Bbbk}\mathcal{O}_{T}$ on $T$.
Let $q:V(-m)_{T}\to\mathcal{F}$ denote the family over $T$ of quotients. We
set the image of $(g,q)$ to be the following family of quotients

${V(-m)_{T}}$${V(-m)_{T}}$${\mathcal{F}.}$$\scriptstyle{g^{-1}(-m)}$$\scriptstyle{q}$
(23)

It is easy to check the above maps are natural in $T\in\mathrm{Sch}/\Bbbk$ and
each defines a group action. They define an algebraic group action
$\mathrm{GL}(V)\curvearrowright\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$.
Moreover, the action by $\mathrm{GL}(V)$ is linear with respect to
$\mathcal{M}$. See for example [9] Section 4.3, where the authors considered
right actions.

We consider the linear action of the subgroup $\mathrm{SL}(V)$. It is easy to
see $R$ is invariant under $\mathrm{SL}(V)$. Let
$\overline{R}\hookrightarrow\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$
be the scheme theoretic image of
$R\to\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$, which is a subscheme
on the closure of $R$ by Lemma 01R8. The linearisation on $\mathcal{M}$ pulls
back to a linearisation $\mathcal{M}|_{\overline{R}}$. We can apply reductive
GIT for the linear action of $\mathrm{SL}(V)$ on $\overline{R}$ with respect
to $\mathcal{M}|_{\overline{R}}$. Note that $\mathcal{M}$ depends on $M\gg
m\gg 0$.

######  Theorem 2.5 ([9] Theorem 4.3.3, Theorem 4.3.4).

For $M\gg m\gg 0$, we have

  * •

GIT (semi)stable points are isomorphism classes
$[(\mathcal{F},q)]\in\overline{R}$ such that $\mathcal{F}$ is
(semi)stable and $q:V(-m)\to\mathcal{E}$ induces an isomorphism $V\cong
H^{0}(B,\mathcal{E}(m))$, i.e.

$\overline{R}^{\mathrm{SL}(V),\mathcal{M},\mathrm{ss}}=R,\quad\overline{R}^{\mathrm{SL}(V),\mathcal{M},\mathrm{s}}=R^{\mathrm{s}};$
(24)

  * •

Two semistable points
$[(\mathcal{F},q)],[(\mathcal{F}^{\prime},q^{\prime})]\in R$ are (GIT)
$S$-equivalent if and only if
$\mathrm{gr}(\mathcal{F})\cong\mathrm{gr}(\mathcal{F}^{\prime})$, where
$\mathrm{gr}(\mathcal{F})$ denotes the associated graded sheaf of the Jordan-
Hölder filtration of $\mathcal{F}$;

  * •

A semistable point $[(\mathcal{F},q)]\in R$ has closed $\mathrm{SL}(V)$-orbit
in $R$ if and only if $\mathcal{F}$ is polystable, i.e. $\mathcal{F}$ is a
direct sum of stable sheaves;

  * •

$\overline{R}/\\!/\mathrm{SL}(V)$ corepresents
$\mathbf{M}_{B/\Bbbk}^{P(t),\mathcal{L},\mathrm{ss}}$ and
$R^{\mathrm{s}}/\mathrm{SL}(V)$ corepresents
$\mathbf{M}_{B/\Bbbk}^{P(t),\mathcal{L},\mathrm{s}}$. Moreover
$R^{\mathrm{s}}/\mathrm{SL}(V)$ is a coarse moduli space of
$\mathbf{M}_{B/\Bbbk}^{P(t),\mathcal{L},\mathrm{s}}$.

######  Remark 2.6.

The coarse moduli space of
$\mathbf{M}_{B/\Bbbk}^{P(t),\mathcal{L},\mathrm{s}}$ has a natural
compactification, which is the corepresentative of
$\mathbf{M}_{B/\Bbbk}^{P(t),\mathcal{L},\mathrm{ss}}$, which is independent of
the GIT problem.

######  Remark 2.7.

If there exists a semistable sheaf $\mathcal{F}$ of Hilbert polynomial $P(t)$
that is not polystable, then
$\mathbf{M}_{B/\Bbbk}^{P(t),\mathcal{L},\mathrm{ss}}$ does not have a coarse
moduli space, since
$\eta:\mathbf{M}_{B/\Bbbk}^{P(t),\mathcal{L},\mathrm{ss}}\to
h_{\overline{R}/\\!/\mathrm{SL}(V)}$ is not bijective on
$\mathrm{Spec}(\Bbbk)$.

####  2.2.2 Moduli of unstable sheaves

Let $\tau$ be a Harder-Narasimhan type of length 2. Let $P_{i}(t)$ for $1\leq
i\leq 2$ be the Hilbert polynomials associated to $\tau$ in the sense that if
$\mathcal{E}$ is a pure sheaf of type $\tau$ on $B$, then $P_{i}(t)$ is the
Hilbert polynomial of
$\mathrm{HN}_{i}(\mathcal{E})/\mathrm{HN}_{i-1}(\mathcal{E})$ for $1\leq i\leq
2$. We have that $P(t):=P_{1}(t)+P_{2}(t)$ is the Hilbert polynomial of
$\mathcal{E}$. Let $V_{i}:=\Bbbk^{P_{i}(m)}$ for $1\leq i\leq 2$ and let
$V:=V_{1}\oplus V_{2}$. Denote $V_{\leq i}:=V_{1}\oplus\cdots\oplus V_{i}$.
Let $\mathrm{P}_{\tau}\subseteq\mathrm{SL}(V)$ be the parabolic subgroup of
$g\in\mathrm{SL}(V)$ stabilising the partial flag $0=V_{\leq 0}\subseteq
V_{\leq 1}\subseteq V_{\leq 2}=V$.

It turns out that it is useful to refine the classification problem according
to the dimension of the space of unipotent endomorphisms. In [11] Definition
7.12 and [7] Definition 4.18, the so-called _$\tau$ -stability_ of pure
sheaves was defined. In [7] Definition 6.1, the so-called _refined Harder-
Narasimhan type_ was defined, which encodes information on the dimension of
the space of unipotent endomorphisms.

Theorem 1.1 in [14] shows that sheaves of fixed Harder-Narasimhan type are
bounded. In the proof of [8] Proposition 6.8, a locally closed subscheme
$Y_{\tau}^{\mathrm{ss}}\subseteq\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$
was described as a parameter space of sheaves of type $\tau$. Closed points in
$Y_{\tau}^{\mathrm{ss}}$ are exactly isomorphism classes
$[(\mathcal{E},q)]\in\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$ such
that each $\mathcal{E}_{i}$ is semistable with Hilbert polynomial $P_{i}(t)$
for $1\leq i\leq 2$, where $\mathcal{E}_{i}:=q\big{(}V_{\leq
i}(-m)\big{)}\big{/}q\big{(}V_{\leq i-1}(-m)\big{)}$. In the same manner, one
can define locally closed subschemes $Y_{\tau}^{\mathrm{s}}$ and $Y_{\tau}$
such that $Y_{\tau}^{\mathrm{s}}\subseteq Y_{\tau}^{\mathrm{ss}}\subseteq
Y_{\tau}\subseteq\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$. Closed
points in $Y_{\tau}\subseteq\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$
are classes $[(\mathcal{E},q)]$ such that $\mathcal{E}_{i}$ has Hilbert
polynomial $P_{i}(t)$, and closed points in $Y_{\tau}^{\mathrm{s}}$ are
classes $[(\mathcal{E},q)]\in Y_{\tau}^{\mathrm{ss}}$ such that
$\mathcal{E}_{i}$ is stable. The locally closed subsets
$Y_{\tau}^{\mathrm{(s)s}}\subseteq Y_{\tau}$ were studied in [8], [10], [11]
and [7], where moduli spaces of _$\tau$ -stable_ sheaves were constructed as
parabolic GIT quotients of $Y_{\tau}^{\mathrm{s}}$.

We will recover the locally closed subschemes
$Y_{\tau}^{\mathrm{(s)s}}\subseteq
Y_{\tau}\subseteq\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$ and actions
of $\mathrm{P}_{\tau}$ on them from a more functorial perspective in this
subsection. From this new perspective, we do not only see closed points but
also view $T$-valued points of $Y^{\mathrm{(s)s}}\subseteq Y_{\tau}$ as
families over $T$ of quotients.

For $Y_{\tau}$, we describe the associated functor of points
$\mathbf{Y}_{\tau}\subseteq\mathbf{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$
as a _sub-functor represented by locally closed subschemes_. Let
$T\in\mathrm{Sch}/\Bbbk$ and let
$[(\mathcal{E},q)]\in\mathbf{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}(T)$ be
an isomorphism class of a quotient

${V(-m)_{T}}$${\mathcal{E}}$${0.}$$\scriptstyle{q}$ (25)

Define the following notation for $1\leq i\leq 2$

$\begin{split}\mathcal{E}_{\leq i}:=&q\big{(}V_{\leq
i}(-m)_{T}\big{)}\subseteq\mathcal{E}\\\ \mathcal{E}_{i}:=&\mathcal{E}_{\leq
i}/\mathcal{E}_{\leq i-1}.\end{split}$ (26)

There is a morphism $q_{i}:V_{i}(-m)_{T}\to\mathcal{E}_{i}$ such that the
following diagram commutes

${0}$${V_{\leq i-1}(-m)_{T}}$${V_{\leq
i}(-m)_{T}}$${V_{i}(-m)_{T}}$${0}$${0}$${\mathcal{E}_{\leq
i-1}}$${\mathcal{E}_{\leq i}}$${\mathcal{E}_{i}}$${0}$$\scriptstyle{q_{\leq
i-1}}$$\scriptstyle{q_{\leq i}}$$\scriptstyle{q_{i}}$ (27)

where $q_{\leq i}:=q|_{V_{\leq i}(-m)_{T}}$ and then the Five lemma shows the
surjectivity of $q_{i}$.

The subset $\mathbf{Y}_{\tau}(T)$ consists of $[(\mathcal{E},q)]$ such that
each $\mathcal{E}_{i}$ is flat over $T$ with Hilbert polynomial $P_{i}(t)$ for
$1\leq i\leq 2$. By Lemma 00HM and the fact that $\mathcal{E}$ is flat over
$T$, we have that each $\mathcal{E}_{\leq i}$ is flat over $T$ if each
$\mathcal{E}_{i}$ is flat over $T$. Since flatness and Hilbert polynomials are
preserved under pullbacks, we have that $\mathbf{Y}_{\tau}$ defines a sub-
functor of $\mathbf{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$.

The following theorem: _flattening stratification_ is important in the
construction of Quot schemes. Here we apply it to show that
$\mathbf{Y}_{\tau}\subseteq\mathbf{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$
is represented by locally closed subschemes.

######  Theorem 2.8 (Theorem 2.1.5, [9]).

Let $X\to T$ be a projective morphism of schemes of finite type over $\Bbbk$.
Let $\mathcal{L}$ be a line bundle on $X$ which is ample relative to $X\to T$.
Let $\mathcal{F}$ be a coherent sheaf on $X$. Then:

  * •

The set of Hilbert polynomials is finite

$\mathcal{P}:=\bigg{\\{}P(t)\in\mathbb{Q}[t]:\begin{matrix}P(t)\textrm{ is the
Hilbert polynomial of }\mathcal{F}\\\ \textrm{at some closed point of
}T\end{matrix}\bigg{\\}};$ (28)

  * •

For each $P(t)\in\mathcal{P}$ the following functor is represented by a
locally closed subscheme $T_{P}\subseteq T$

$\mathbf{Flat}_{\mathcal{F}/X/T}^{P(t),\mathcal{L}}:(\mathrm{Sch}/T)^{\mathrm{op}}\to\mathrm{Set},\quad
T^{\prime}\mapsto\begin{cases}\\{\mathrm{pt}\\},&\begin{matrix}\mathcal{F}_{T^{\prime}}\textrm{
is flat over }T^{\prime}\\\ \textrm{ with Hilbert polynomial
}P(t)\end{matrix}\\\ \emptyset,&\textrm{otherwise}\end{cases}$ (29)

where $\mathrm{Sch}/T$ is the category of schemes of finite type over $T$, and
$\mathcal{F}_{T^{\prime}}$ is the pullback of $\mathcal{F}$ along
$T^{\prime}\to T$;

  * •

The morphism $\coprod_{P\in\mathcal{P}}T_{P}\to T$ is bijective on points.

######  Lemma 2.9.

$\mathbf{Y}_{\tau}\subseteq\mathbf{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$
is representable by locally closed subschemes.

###### Proof.

Let $T\in\mathrm{Sch}/\Bbbk$ and consider the following diagram in
$\mathrm{PSh}(\mathrm{Sch}/\Bbbk)$

${h_{T}}$${\mathbf{Y}_{\tau}}$${\mathbf{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}}$$\scriptstyle{\xi}$$\scriptstyle{\subseteq}$
(30)

where $\xi:h_{T}\to\mathbf{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$ is
identified with a class
$[(\mathcal{E},q)]\in\mathbf{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}(T)$ by
the Yoneda lemma.

Recall $\mathcal{E}_{\leq i}:=q\big{(}V_{\leq i}(-m)_{T}\big{)}$ and
$\mathcal{E}_{i}:=\mathcal{E}_{\leq i}/\mathcal{E}_{\leq i-1}$. For each
$1\leq i\leq 2$ consider the functor
$\mathbf{Flat}_{\mathcal{E}_{i}/B_{T}/T}^{P_{i}(t),\mathcal{L}_{T}}$ defined
in Theorem 2.8, which is represented by a locally closed subscheme
$T_{\xi,P_{i}}\subseteq T$. Let

$T_{\xi}:=T_{\xi,P_{1}}\times_{T}T_{\xi,P_{2}}$ (31)

which is a locally closed subscheme of $T$.

For $T^{\prime}\to T$ in $\mathrm{Sch}/\Bbbk$, denote
$\mathcal{E}|_{T^{\prime}}$ the pullback of $\mathcal{E}$ along
$B_{T^{\prime}}\to B_{T}$, and similar for $\mathcal{E}_{i},\mathcal{E}_{\leq
i}$. We have

$\begin{split}&T^{\prime}\to T\textrm{ factors through }T^{\prime}\to
T_{\xi}\to T\\\ \textrm{if and only if}\quad&T^{\prime}\to T\textrm{ factors
though }T_{\xi,P_{i}}\textrm{ for each }1\leq i\leq 2\\\ \textrm{if and only
if}\quad&\begin{cases}\mathcal{E}_{i}|_{T^{\prime}}\textrm{ is flat over
}T^{\prime}\\\ \textrm{with Hilbert polynomial }P_{i}(t)\end{cases}\textrm{
for }1\leq i\leq 2\\\ \textrm{if and only if}\quad&(T^{\prime}\to
T)^{*}\xi\in\mathbf{Y}_{\tau}(T^{\prime}).\end{split}$ (32)

The above equivalences show the following diagram is Cartesian

${h_{T_{\xi}}}$${h_{T}}$${\mathbf{Y}_{\tau}}$${\mathbf{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}}$$\scriptstyle{\subseteq}$$\scriptstyle{\xi}$$\scriptstyle{\subseteq}$
(33)

which proves the representability of
$\mathbf{Y}_{\tau}\subseteq\mathbf{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$
by locally closed subschemes. ∎

Since $\mathbf{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$ is represented by
$\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$, by Lemma 2.9, there is a
locally closed subscheme
$Y_{\tau}\subseteq\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$
representing $\mathbf{Y}_{\tau}$.

Consider
$\prod_{i=1}^{2}\mathrm{Quot}_{V_{i}(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}$,
where the fibre product is over $\Bbbk$. There is a morphism
$\prod_{i=1}^{2}\mathrm{Quot}_{V_{i}(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}\to\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$.
From their functors of points, we easily see that this morphism is a
monomorphism, and it is proper by Lemma 01W6, thus a closed immersion by Lemma
04XV. The closed immersion
$\prod_{i=1}^{2}\mathrm{Quot}_{V_{i}(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}\hookrightarrow\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$
factors through $Y_{\tau}$, which is easily seen from the associated functors
of points. There is a retraction
$p_{\tau}:Y_{\tau}\to\prod_{i=1}^{2}\mathrm{Quot}_{V_{i}(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}$
such that the following diagram commutes

${\prod_{i=1}^{2}\mathrm{Quot}_{V_{i}(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}}$${Y_{\tau}}$${\prod_{i=1}^{2}\mathrm{Quot}_{V_{i}(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}}$$\scriptstyle{\subseteq}$$\scriptstyle{=}$$\scriptstyle{p_{\tau}}$
(34)

where $p_{\tau}$ sends an isomorphism class $[(\mathcal{E},q)]$ of families
over $T\in\mathrm{Sch}/\Bbbk$ to
$\Big{[}\Big{(}\bigoplus_{i=1}^{2}\mathcal{E}_{i},\bigoplus_{i=1}^{2}q_{i}\Big{)}\Big{]}$,
where $\mathcal{E}_{i}$ and $q_{i}$ are defined in equation (26) and equation
(27).

Consider $(\mathbb{G}_{m})^{2}\to\mathrm{GL}(V)$ such that
$(t_{1},t_{2})\mapsto t_{1}1_{V_{1}}\oplus t_{2}1_{V_{2}}\in\mathrm{GL}(V)$.
Then $(\mathbb{G}_{m})^{2}$ acts on $Y_{\tau}$. With this
$(\mathbb{G}_{m})^{2}$-action, we have that
$\prod_{i=1}^{2}\mathrm{Quot}_{V_{i}(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}$ is
the _fixed point subscheme_ $(Y_{\tau})^{(\mathbb{G}_{m})^{2}}\hookrightarrow
Y_{\tau}$ in the sense of [4]. It is easy to see if we consider
$\lambda:\mathbb{G}_{m}\to(\mathbb{G}_{m})^{2}$ given by
$t\mapsto(t^{w_{1}},t^{w_{2}})$ such that $w_{1}>w_{2}$, then the fixed point
subscheme of the action of $\lambda$ is also
$\prod_{i=1}^{2}\mathrm{Quot}_{V_{i}(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}$

$\prod_{i=1}^{2}\mathrm{Quot}_{V_{i}(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}\cong(Y_{\tau})^{(\mathbb{G}_{m})^{2}}=(Y_{\tau})^{\lambda}.$
(35)

We then describe the open subschemes $Y_{\tau}^{\mathrm{s}}\subseteq
Y_{\tau}^{\mathrm{ss}}\subseteq Y_{\tau}$. For $1\leq i\leq 2$, let
$R_{i}^{\mathrm{(s)s}}\subseteq\mathrm{Quot}_{V_{i}(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}$
be the open subscheme whose closed points are $[(\mathcal{E}_{i},q_{i})]$ such
that $\mathcal{E}_{i}$ is a (semi)stable pure sheaf on $B$ and
$q:V_{i}(-m)\to\mathcal{E}_{i}$ induces an isomorphism $V_{i}\cong
H^{0}(B,\mathcal{E}_{i}(m))$ (cf. Theorem 2.5). We define
$Y_{\tau}^{\mathrm{(s)s}}$ as the fibre product indicated in the diagram below

${Y_{\tau}^{\mathrm{(s)s}}}$${Y_{\tau}}$${\prod_{i=1}^{2}R_{i}^{\mathrm{(s)s}}}$${\prod_{i=1}^{2}\mathrm{Quot}_{V_{i}(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}.}$${\lrcorner}$$\scriptstyle{\subseteq}$$\scriptstyle{p_{\tau}^{\mathrm{(s)s}}}$$\scriptstyle{p_{\tau}}$$\scriptstyle{\subseteq}$
(36)

We have that $Y_{\tau}^{(s)s}\subseteq Y_{\tau}$ is open. So
$p_{\tau}^{\mathrm{(s)s}}:Y_{\tau}^{\mathrm{(s)s}}\to\prod_{i=1}^{2}R_{i}^{\mathrm{(s)s}}$
are also retractions as base changes of $p_{\tau}$. In particular, $p_{\tau}$
and $p_{\tau}^{\mathrm{(s)s}}$ are surjective.

There is a $\mathrm{P}_{\tau}$-action on $Y_{\tau}$ induced from the
$\mathrm{SL}(V)$-action on $\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$.
Similarly there are $\mathrm{P}_{\tau}$-actions on
$Y_{\tau}^{\mathrm{s}}\subseteq Y_{\tau}^{\mathrm{ss}}$. These
$\mathrm{P}_{\tau}$-actions are linearised with respect to the restrictions of
$\mathcal{M}$ on $\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$, where
$\mathcal{M}$ is the ample line bundle described in (21).

In [10] and [11], Jackson considered the induced reduced subscheme structure
on $Y_{\tau}^{\mathrm{s}}$ for a Harder-Narasimhan type $\tau$ of length 2,
and stratify this reduced subscheme further according to the dimension of the
stabiliser in the unipotent radical of $\mathrm{P}_{\tau}$. Non-reductive GIT
could be applied to each stratum and the following theorem was proved.

######  Definition 2.10 (cf. [11] Definition 7.1.2, [7] Definition 6.1).

Let $B$ be a connected projective scheme over $\Bbbk$ with an ample line
bundle $\mathcal{L}$. Let $\tau$ be a Harder-Narasimhan type of length 2. Let
$\delta\in\mathbb{N}$. A pure sheaf $\mathcal{E}$ on $B$ is called
_$(\tau,\delta)$ -stable_ if:

  * •

$\mathcal{E}$ has Harder-Narasimhan type $\tau$;

  * •

if $\\{\mathcal{E}_{\leq i}\\}_{i=0}^{2}$ is the Harder-Narasimhan filtration,
then $\mathcal{E}_{i}:=\mathcal{E}_{\leq i}/\mathcal{E}_{\leq i-1}$ is stable
for $i=1,2$;

  * •

$\mathcal{E}\not\cong\mathcal{E}_{1}\oplus\mathcal{E}_{2}$;

  * •

$\dim\mathrm{Hom}_{\mathcal{O}_{B}}(\mathcal{E}_{2},\mathcal{E}_{1})=\delta$.

######  Theorem 2.11 ([11] Theorem 7.1.3).

Let $B$ be a connected projective scheme over $\Bbbk$ with an ample line
bundle $\mathcal{L}$. Let $\tau$ be a Harder-Narasimhan type of length 2 and
let $\delta\in\mathbb{N}$. Then there exists a quasi-projective moduli space
$M_{\tau,\delta}$ of $(\tau,\delta)$-stable sheaves on $B$, in the sense that
closed points in $M_{\tau,\delta}$ are exactly isomorphism classes of
$(\tau,\delta)$-stable sheaves.

######  Remark 2.12.

The moduli spaces $M_{\tau,\delta}$ above are reduced schemes since they were
constructed by the classic U-hat theorem (cf. Theorem 2.20, [2]), which only
produces reduced quotient schemes.

The non-reductive GIT problem considered in [11] to prove Theorem 2.11 was for
the linearisation

$\mathrm{P}_{\tau}\curvearrowright\big{(}\overline{Y_{\tau}^{\mathrm{s}}},\;\mathcal{M}\big{|}_{\overline{Y_{\tau}^{\mathrm{s}}}}\big{)}$
(37)

where
$\overline{Y_{\tau}^{\mathrm{s}}}\subseteq\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$
is the scheme theoretic closure of $Y_{\tau}^{\mathrm{s}}$ in
$\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P(t),\mathcal{L}}$. In Appendix A, we study
the boundary of
$Y_{\tau}^{\mathrm{s}}\subseteq\overline{Y_{\tau}^{\mathrm{s}}}$ and
Proposition A.2 shows that when applying our main theorem in non-reductive GIT
(Theorem 3.33).

For each $\delta\in\mathbb{N}$, there exists a locally closed subset
$Y_{\tau,\delta}^{\mathrm{s}}$ (See equation (187) for its definition), whose
closed points are $[(\mathcal{E},q)]$ such that $\mathcal{E}$ is
$(\tau,\alpha)$-stable. Each of $Y_{\tau,\delta}^{\mathrm{s}}$ has a quasi-
projective universal geometric quotient by $\mathrm{P}_{\tau}$

$Y_{\tau,\delta}^{\mathrm{s}}\to
Y_{\tau,\delta}^{\mathrm{s}}/\mathrm{P}_{\tau}.$ (38)

Note that in [11], only closed points of
$Y_{\tau,\delta}^{\mathrm{s}}\subseteq
Y_{\tau}^{\mathrm{s}}\subseteq\overline{Y_{\tau}^{\mathrm{s}}}$ were
described, and scheme structures were implicitly the reduced ones. However by
[3] Remark 7.8, even for rank 2 unstable bundles over curves, the moduli space
need not be reduced.

One key fact peculiar to length two is the following, which fails for higher
lengths.

######  Lemma 2.13 ([11] Corollary 7.2.6).

For any closed point $x\in Y_{\tau}^{\mathrm{ss}}$, we have

$\dim\mathrm{Stab}_{\mathrm{U}_{\tau}}(x)=\dim\mathrm{Stab}_{\mathrm{U}_{\tau}}(p_{\tau}(x))$
(39)

where
$p_{\tau}:Y_{\tau}\to\prod_{i=1}^{2}\mathrm{Quot}_{V(-m)/B/\Bbbk}^{P_{i}(t),\mathcal{L}}$
is the retraction and $\mathrm{U}_{\tau}\subseteq\mathrm{P}_{\tau}$ is the
unipotent radical.

We will consider the non-reduced version of the GIT problem above. This
requires:

  * •

an extension of non-reductive GIT developed in [2], [1] to linear actions on
non-reduced schemes;

  * •

a scheme theoretic description of $Y_{\tau,\delta}^{\mathrm{s}}$.

We will address these in the following two sections respectively.

##  3 Non-reductive GIT for graded additive groups

Let $U$ be an additive group, i.e. $U\cong(\mathbb{G}_{a})^{r}$ for $r:=\dim
U$. Let $\lambda:\mathbb{G}_{m}\to\mathrm{Aut}(U)$ be a 1-parameter subgroup
acting on $U$ by conjugation such that the induced representation of
$\mathbb{G}_{m}$ on $\mathfrak{u}:=\mathrm{Lie}(U)$ has a unique positive
weight. Let $\hat{U}:=U\rtimes_{\lambda}\mathbb{G}_{m}$ be the semi-direct
product of $U$ and the 1PS $\lambda$. We will consider group actions by
$\hat{U}$.

###  3.1 Some definitions for group actions

All schemes are Noetherian over $\Bbbk$. Let $G$ be an affine algebraic group.
Let $X$ be a scheme acted on by $G$ via the action morphism $\sigma:G\times
X\to X$. Let $\mathfrak{g}^{*}$ be the dual of the Lie algebra of $G$.

####  3.1.1 Infinitesimal stabilisers of scheme-valued points

######  Definition 3.1.

Let $\sigma:G\times X\to X$ be an action. Let $t:T\to X$ be a morphism in
$\mathrm{Sch}/\Bbbk$. The _stabiliser_ of the $T$-point $t$, denoted by
$\mathrm{Stab}_{G}(t)$ or $\mathrm{Stab}_{G}(T)$, is defined as the following
fibre product

${\mathrm{Stab}_{G}(t)}$${T}$${G\times T}$${X\times
T.}$${\lrcorner}$$\scriptstyle{(t,1_{T})}$$\scriptstyle{\big{(}\sigma\circ(\mathrm{pr}_{0},t\circ\mathrm{pr}_{1}),\;\mathrm{pr}_{1}\big{)}}$
(40)

The _infinitesimal stabiliser_ of the $T$-point $t$, denoted by
$\mathrm{Stab}_{\mathfrak{g}}(t)$ or $\mathrm{Stab}_{\mathfrak{g}}(T)$, is
defined as the normal sheaf of the closed immersion
$e^{\prime\prime}:T\hookrightarrow\mathrm{Stab}_{G}(t)$

${T}$${\mathrm{Stab}_{G}(t)}$${T}$${G\times T}$${X\times
T}$$\scriptstyle{1_{T}}$$\scriptstyle{e^{\prime\prime}}$$\scriptstyle{e^{\prime}}$${\lrcorner}$$\scriptstyle{(t,1_{T})}$$\scriptstyle{\big{(}\sigma\circ(\mathrm{pr}_{0},t\circ\mathrm{pr}_{1}),\;\mathrm{pr}_{1}\big{)}}$
(41)

induced by the closed immersion $e^{\prime}:T\hookrightarrow G\times T$ at the
identity $e:\mathrm{Spec}(\Bbbk)\hookrightarrow G$

$\mathrm{Stab}_{\mathfrak{g}}(t):=\mathcal{N}_{e^{\prime\prime}}\equiv\mathcal{H}om_{\mathcal{O}_{T}}\big{(}\mathcal{C}_{T/\mathrm{Stab}_{G}(t)},\mathcal{O}_{T}\big{)}.$
(42)

Consider the following diagram for a morphism $t:T\to X$ in
$\mathrm{Sch}/\Bbbk$

${T}$${\mathrm{Stab}_{G}(t)}$${T}$${G\times T}$${T}$${X\times
T}$$\scriptstyle{e^{\prime\prime}}$$\scriptstyle{e^{\prime}}$$\scriptstyle{\big{(}\sigma\circ(\mathrm{pr_{G}},t\circ\mathrm{pr}_{T}),\mathrm{pr}_{T}\big{)}}$$\scriptstyle{(t,1_{T})}$
(43)

where horizontal morphisms are all immersions. By Lemma 01R3, there is a chain
of conormal sheaves on $T$

${\mathcal{C}_{(t,1_{T})}}$${\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{T}}$${\mathcal{C}_{T/\mathrm{Stab}_{G}(t)}}$${0.}$
(44)

which is exact since $\mathrm{Stab}_{G}(t)\to G\times T$ is an immersion and
$\mathrm{Stab}_{G}(t)\cong(G\times T)\times_{X\times T}T$. The following
diagram is Cartesian

${T}$${X\times T}$${X}$${X\times
X}$${\lrcorner}$$\scriptstyle{(t,1_{T})}$$\scriptstyle{t}$$\scriptstyle{1_{X}\times
t}$$\scriptstyle{\delta}$ (45)

where $\delta:X\to X\times X$ is the diagonal morphism, and horizontal
morphisms are immersions. By Lemma 01R4, there is a surjective morphism of
conormal sheaves

${t^{*}\Omega_{X/\Bbbk}}$${\mathcal{C}_{(t,1_{T})}}$${0.}$ (46)

Then the following sequence is exact

${t^{*}\Omega_{X/\Bbbk}}$${\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{T}}$${\mathcal{C}_{T/\mathrm{Stab}_{G}(t)}}$${0.}$
(47)

######  Definition 3.2.

Let $\sigma:G\times X\to X$ be an action. Choose $t:T\to X$ to be the identity
morphism $1_{X}:X\to X$. We call
$\phi:\Omega_{X/\Bbbk}\to\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}$
constructed above the _infinitesimal action of $\mathfrak{g}$ on $X$_.

######  Lemma 3.3.

Let $f:X_{1}\to X_{2}$ be an immersion of schemes. Let $G$ act on
$X_{1},X_{2}$ such that $f$ is $G$-invariant. Let
$\phi_{i}:\Omega_{X_{i}/\Bbbk}\to\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X_{i}}$
be the infinitesimal action of $\mathfrak{g}$ on $X_{i}$ for $i=1,2$. Then we
have $\mathrm{coker}(\phi_{1})\cong f^{*}\mathrm{coker}(\phi_{2})$ fitting
into the diagram

${f^{*}\Omega_{X_{2}/\Bbbk}}$${\mathfrak{g}^{*}\otimes_{\Bbbk}f^{*}\mathcal{O}_{X_{2}}}$${f^{*}\mathrm{coker}(\phi_{2})}$${0}$${\Omega_{X_{1}/\Bbbk}}$${\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X_{1}}}$${\mathrm{coker}(\phi_{1})}$${0.}$$\scriptstyle{f^{*}\phi_{2}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\phi_{1}}$
(48)

###### Proof.

The morphism $f$ is $G$-invariant, so the left square commutes. The left
square induces a morphism
$f^{*}\mathrm{coker}(\phi_{2})\to\mathrm{coker}(\phi_{1})$. Since $f$ is an
immersion, we have that $f^{*}\Omega_{X_{2}/\Bbbk}\to\Omega_{X_{1}/\Bbbk}$ is
surjective. Therefore
$f^{*}\mathrm{coker}(\phi_{2})\to\mathrm{coker}(\phi_{1})$ is an isomorphism
by the Five lemma. ∎

######  Lemma 3.4.

Let $G$ act on $X$. In the category $\mathrm{Sch}/X$, we have

  * (1)

The formation of $\mathrm{Stab}_{G}(t)$ commutes with arbitrary base change,
in the sense that if $T^{\prime}\to T$ is a morphism in $\mathrm{Sch}/X$
between $t^{\prime}:T^{\prime}\to X$ and $t:T\to X$, then the following are
Cartesian

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
8.69186pt\hfil&\hfil\hskip 46.97148pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{20.69185pt}{-15.20418pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-27.46854pt}{-24.26387pt}\pgfsys@lineto{-3.49852pt}{-24.26387pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (49)

  * (2)

The formation of $\mathrm{Stab}_{\mathfrak{g}}(t)$ commutes with flat base
change, in the sense that if $T^{\prime}\to T$ is a flat morphism in
$\mathrm{Sch}/X$ between $t^{\prime}:T^{\prime}\to X$ and $t:T\to X$, then

$(T^{\prime}\to
T)^{*}\mathrm{Stab}_{\mathfrak{g}}(t)\cong\mathrm{Stab}_{\mathfrak{g}}(t^{\prime}).$
(50)

  * (3)

The formation of sequence (47) commutes with arbitrary base change in the
sense that if $T^{\prime}\to T$ is a morphism in $\mathrm{Sch}/X$ between
$t^{\prime}:T^{\prime}\to X$ and $t:T\to X$, then the sequence

${t^{*}\Omega_{X/\Bbbk}}$${\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{T}}$${\mathcal{C}_{T/\mathrm{Stab}_{G}(t)}}$${0}$
(51)

pulls back along $T^{\prime}\to T$ to the sequence

${(t^{\prime})^{*}\Omega_{X/\Bbbk}}$${\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{T^{\prime}}}$${\mathcal{C}_{T^{\prime}/\mathrm{Stab}_{G}(t^{\prime})}}$${0.}$
(52)

Therefore sequence (47) is the cokernel sequence of $t^{*}\phi$ for
$\phi:\Omega_{X/\Bbbk}\to\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}$.

###### Proof.

It is routine to check $(1)$ and $(3)$. Note that in $(2)$ we need flatness
since
$\mathrm{Stab}_{\mathfrak{g}}(t)=\mathcal{H}om_{\mathcal{O}_{T}}\big{(}\mathcal{C}_{T/\mathrm{Stab}_{G}(t)},\mathcal{O}_{T}\big{)}$
and the formation of dual sheaves commutes with flat base change by Lemma
0C6I. ∎

######  Lemma 3.5.

Let $G$ act on $X$. Let
$\phi:\Omega_{X/\Bbbk}\to\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}$ be
the infinitesimal action. Let $t:T\to X$ be a $T$-point. Let
$\delta\in\mathbb{N}$. The following are equivalent:

  * (1)

$t^{*}\mathrm{coker}(\phi)$ is locally free of rank $\delta$ on $T$;

  * (2)

$\mathrm{Stab}_{\mathfrak{g}}(t^{\prime})$ is locally free of rank $\delta$ on
$T^{\prime}$ for all $T^{\prime}\to T$, where $t^{\prime}$ is the composition
$t^{\prime}:T^{\prime}\to T\xrightarrow{t}X$;

  * (3)

the same in $(2)$ with $T^{\prime}\to T$ ranging over all closed subschemes of
$T$.

###### Proof.

Assume $(1)$. Then $(t^{\prime})^{*}\mathrm{coker}(\phi)$ is locally free of
rank $\delta$. Recall
$\mathrm{Stab}_{\mathfrak{g}}(t^{\prime})=\mathcal{H}om_{\mathcal{O}_{T^{\prime}}}\big{(}(t^{\prime})^{*}\mathrm{coker}(\phi),\mathcal{O}_{T^{\prime}}\big{)}$,
which is locally free of rank $\delta$ on $T^{\prime}$. This proves
$(1)\implies(2)$.

Obviously $(2)\implies(3)$.

Assume $(3)$. For $k\in\mathbb{N}$, let $Z_{k}\subseteq T$ denote the closed
subscheme associated to the $k$th Fitting ideal of
$t^{*}\mathrm{coker}(\phi)$. For $S_{k}:=Z_{k-1}\setminus Z_{k}$, we have
$\mathrm{coker}(\phi)|_{S_{k}}:=(S_{k}\to T)^{*}t^{*}\mathrm{coker}(\phi)$ is
locally free of rank $k$ by Lemma 05P8. Then
$\mathrm{Stab}_{\mathfrak{g}}(S_{k})=\mathcal{H}om_{\mathcal{O}_{S_{k}}}\big{(}\mathrm{coker}(\phi)|_{S_{k}},\mathcal{O}_{S_{k}}\big{)}$
is locally free of rank $k$ on $S_{k}$. Condition $(3)$ implies that
$\mathrm{Stab}_{\mathfrak{g}}(Z_{k-1})$ is locally free of rank $\delta$ on
$Z_{k-1}$. Since $S_{k}\to Z_{k-1}$ is an open immersion, we have
$\mathrm{Stab}_{\mathfrak{g}}(Z_{k-1})|_{S_{k}}\cong\mathrm{Stab}_{\mathfrak{g}}(S_{k})$
by Lemma 3.4. Therefore if $k\neq\delta$, then $S_{k}=\emptyset$. Then
$Z_{\delta-1}=T$ and $Z_{\delta}=\emptyset$.

Condition $(3)$ implies that $\mathrm{Stab}_{\mathfrak{g}}(T)$ is locally free
of rank $\delta$ on $T$. Lemma 05P8 implies that $t^{*}\mathrm{coker}(\phi)$
is locally generated by $\leq\delta$ sections. Let $W\subseteq T$ be an affine
open subset such that $\mathrm{Stab}_{\mathfrak{g}}(T)|_{W}$ is free of rank
$\delta$ and such that there exists an exact sequence of
$\mathcal{O}_{W}$-modules for some $N\in\mathbb{N}$

${\mathcal{O}_{W}^{\oplus
N}}$${\mathcal{O}_{W}^{\oplus\delta}}$${\mathrm{coker}(\phi)|_{W}}$${0}$$\scriptstyle{\eta}$$\scriptstyle{\zeta}$
(53)

where $\mathrm{coker}(\phi)|_{W}:=t^{*}\mathrm{coker}(\phi)|_{W}$. The
sequence of dual $\mathcal{O}_{W}$-modules is exact

${0}$${\mathrm{Stab}_{\mathfrak{g}}(W)}$${\mathcal{O}_{W}^{\oplus\delta}}$${\mathcal{O}_{W}^{\oplus
N}.}$$\scriptstyle{\zeta^{*}}$$\scriptstyle{\eta^{*}}$ (54)

We have that $\mathrm{Stab}_{\mathfrak{g}}(W)$ is free of rank $\delta$ since
$\mathrm{Stab}_{\mathfrak{g}}(W)\cong\mathrm{Stab}_{\mathfrak{g}}(T)|_{W}$ by
Lemma 3.4. It is well known that $\det(\zeta^{*})\in\mathcal{O}_{W}(W)$ is a
non-zerodivisor when $\zeta^{*}$ is an injective map of finite free modules of
the same rank $\delta$. Moreover, $\mathrm{coker}(\zeta^{*})$ is annihilated
by $\det(\zeta^{*})$ and
$\mathrm{coker}(\zeta^{*})\cong\mathrm{im}(\eta^{*})\subseteq\mathcal{O}_{W}^{\oplus
N}$ is contained in a torsion free module. Therefore
$\mathrm{coker}(\zeta^{*})\cong\mathrm{im}(\eta^{*})=0$, i.e. $\eta=0$, which
implies that
$\zeta:\mathcal{O}_{W}^{\oplus\delta}\to\mathrm{coker}(\phi)|_{W}$ is an
isomorphism. This proves $(1)$. ∎

For the infinitesimal action
$\phi:\Omega_{X/\Bbbk}\to\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}$, let
$\mathrm{Fit}_{k}(\phi)\subseteq\mathcal{O}_{X}$ denote the $k$th Fitting
ideal of $\mathrm{coker}(\phi)$.

######  Lemma 3.6.

Let $G$ act on $X$. Let
$\phi:\Omega_{X/\Bbbk}\to\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}$ be
the infinitesimal action. Let $\mathbb{K}$ be a field. Let
$x:\mathrm{Spec}(\mathbb{K})\to X$ be a $\mathbb{K}$-point. Let
$k\in\mathbb{Z}$. The following are equivalent:

  * (1)

$\dim_{\mathbb{K}}\mathrm{Stab}_{\mathfrak{g}}(x)>k$;

  * (2)

$\mathrm{Fit}_{k}(\phi)\subseteq\ker(x^{\sharp})$, where
$x^{\sharp}:\mathcal{O}_{X}\to x_{*}(\mathbb{K})$ is the morphism of sheaves
of rings associated to $x$.

###### Proof.

Let $r:=\dim_{\Bbbk}\mathfrak{g}$. When $k\geq r$ or $k\leq-1$, both $(1)$ and
$(2)$ are false or both are true. We only need to consider when $0\leq k<r$.

Let $\mathrm{Spec}(A)\subseteq X$ be an open neighbourhood of $x\in X$. Let
$x_{A}:\mathrm{Spec}(\mathbb{K})\to\mathrm{Spec}(A)$ be the morphism when $x$
is viewed as a $\mathbb{K}$-point of $\mathrm{Spec}(A)$. On
$\mathrm{Spec}(A)$, the infinitesimal action
$\phi:\Omega_{X/\Bbbk}\to\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}$
corresponds to a map of $A$-modules

$\phi_{A}:\Omega_{A/\Bbbk}\to\mathfrak{g}^{*}\otimes_{\Bbbk}A.$ (55)

Choose a surjective map

$\alpha:A^{\oplus J}\to\Omega_{A/\Bbbk}$ (56)

for an index set $J$. Choose a basis $(u_{1},\cdots,u_{r})$ of
$\mathfrak{g}^{*}$. Let $M=(m_{ij})_{\begin{subarray}{c}1\leq i\leq r\\\ j\in
J\end{subarray}}$ be the matrix with entries $m_{ij}\in A$ such that

$(\phi_{A}\circ\alpha)(e_{j})=\sum_{i=1}^{r}u_{i}\otimes
m_{ij},\quad\textrm{for all }j\in J.$ (57)

Then the ideal $\mathrm{Fit}_{k}(\phi_{A})\subseteq A$ is generated by
$(r-k)\times(r-k)$-minors of $M$.

Let $x_{A}^{\sharp}:A\to\mathbb{K}$ be the ring map corresponding to the point
$x_{A}:\mathrm{Spec}(\mathbb{K})\to\mathrm{Spec}(A)$. The base change of
$\phi_{A}\circ\alpha$ along $x_{A}^{\sharp}:A\to\mathbb{K}$ is

$x_{A}^{*}(\phi_{A}\circ\alpha):\mathbb{K}^{\oplus
J}\to\mathfrak{g}^{*}\otimes_{\Bbbk}\mathbb{K}$ (58)

where $\Bbbk\to\mathbb{K}$ corresponds to
$\mathrm{Spec}(\mathbb{K})\xrightarrow{x}X\to\mathrm{Spec}(\Bbbk)$. Then the
linear map $x_{A}^{*}(\phi_{A}\circ\alpha)$ is represented by the matrix
$x_{A}^{\sharp}(M):=\big{(}x_{A}^{\sharp}(m_{ij})\big{)}$ with entries in
$\mathbb{K}$. Note that $\xi\in\mathfrak{g}\otimes_{\Bbbk}\mathbb{K}$
stabilises $x$ if and only if
$\mathrm{im}\big{(}x_{A}^{*}(\phi_{A}\circ\alpha)\big{)}\subseteq\xi^{\perp}$.

We have

$\begin{split}(1)\quad\iff&\quad\textrm{there exist independent
}\xi_{0},\cdots,\xi_{k}\in\mathfrak{g}\otimes_{\Bbbk}\mathbb{K}\textrm{
stabilising }x\\\ \iff&\quad\textrm{there exist independent
}\xi_{0},\cdots,\xi_{k}\in\mathfrak{g}\otimes_{\Bbbk}\mathbb{K}\\\
&\qquad\textrm{such that
}\mathrm{im}\big{(}x_{A}^{*}(\phi_{A}\circ\alpha)\big{)}\subseteq(\xi_{0},\cdots,\xi_{k})^{\perp}\\\
\iff&\quad\dim_{\mathbb{K}}\mathrm{im}\big{(}x_{A}^{*}(\phi_{A}\circ\alpha)\big{)}<r-k\\\
\iff&\quad\mathrm{rank}(x_{A}^{\sharp}M)<r-k\\\ \iff&\quad\textrm{any
}(r-k)\times(r-k)\textrm{-minor of }x_{A}^{\sharp}M\textrm{ vanishes}\\\
\iff&\quad\textrm{any }(r-k)\times(r-k)\textrm{-minor of }M\textrm{ is in
}\ker(x_{A}^{\sharp})\\\
\iff&\quad\mathrm{Fit}_{k}(\phi_{A})\subseteq\ker(x_{A}^{\sharp})\\\
\iff&\quad(2)\end{split}$ (59)

∎

######  Remark 3.7.

If $\mathrm{coker}(\phi)$ is locally free of rank $k$ on $X$, then
$\mathrm{Fit}_{k-1}(\phi)=0$ and $\mathrm{Fit}_{k}(\phi)=\mathcal{O}_{X}$. By
Lemma 3.6, this implies $\dim_{\mathbb{K}}\mathrm{Stab}_{\mathfrak{g}}(x)=k$
for any field $\mathbb{K}$ and $\mathbb{K}$-point $x\in X$. Conversely, if
$\dim_{\mathbb{K}}\mathrm{Stab}_{\mathfrak{g}}(x)=k$ for any $\mathbb{K}$ and
$\mathbb{K}$-point $x\in X$, then we only have
$\mathrm{Fit}_{k-1}(\phi)\subseteq\mathrm{nil}(X)$ and
$\mathrm{Fit}_{k}(\phi)=\mathcal{O}_{X}$, where
$\mathrm{nil}(X)\subseteq\mathcal{O}_{X}$ is the nilpotent radical.

####  3.1.2 Scheme-valued infinitesimal transformations

Let $T$ be a scheme over $\Bbbk$. Recall that a $T$-valued point of $X$ is a
morphism $t:T\to X$. We will define _$T$ -valued infinitesimal transformations
in $G$_ and when a $T$-valued infinitesimal transformation stabilises a
$T$-valued point.

Let $\Bbbk[\epsilon]:=\Bbbk[t]/t^{2}$ be the algebra of dual numbers. For
$T\in\mathrm{Sch}/\Bbbk$, let
$T[\epsilon]:=T\times_{\Bbbk}\mathrm{Spec}(\Bbbk[\epsilon])$. The underlying
topological spaces of $T$ and $T[\epsilon]$ are the same. We can think of
$T[\epsilon]$ as a scheme on the topological space $|T|$ with the structure
sheaf

$\mathcal{O}_{T[\epsilon]}:=\mathcal{O}_{T}\otimes_{\Bbbk}\Bbbk[\epsilon]\cong\mathcal{O}_{T}\oplus\epsilon\mathcal{O}_{T}.$
(60)

There is a natural closed immersion $T\hookrightarrow T[\epsilon]$,
corresponding to
$\mathcal{O}_{T[\epsilon]}\xrightarrow{\epsilon=0}\mathcal{O}_{T}$.

######  Definition 3.8.

Let $G$ be an affine algebraic group over $\Bbbk$. A _$T$ -valued
infinitesimal transformation in $G$_ is a morphism $\rho:T[\epsilon]\to G$
satisfying the diagram

${T}$${T[\epsilon]}$${\mathrm{Spec}(\Bbbk)}$${G.}$$\scriptstyle{\rho}$$\scriptstyle{e}$
(61)

######  Definition 3.9.

Let $G$ act on $X$. Let $t:T\to X$ be a $T$-valued point of $X$, and let
$\rho:T[\epsilon]\to G$ be a $T$-valued infinitesimal transformation in $G$.
We say that $\rho$ _stabilises_ $t$ if the following diagram commutes

${T[\epsilon]}$${G\times
X}$${T}$${X.}$$\scriptstyle{(\rho,t\circ\mathrm{pr})}$$\scriptstyle{\mathrm{pr}}$$\scriptstyle{\sigma}$$\scriptstyle{t}$
(62)

######  Remark 3.10.

If we take $T=\mathrm{Spec}(\Bbbk)$, then $T$-valued points of $X$ are
$\Bbbk$-points, and $T$-valued infinitesimal transformations in $G$ are
vectors in the Lie algebra $\mathfrak{g}$. For a $T$-valued point $x\in X$ and
a $T$-valued infinitesimal transformation $\xi\in\mathfrak{g}$, the definition
of when $\xi$ stabilises $x$ in Definition 3.9 is equivalent to the
requirement that the pullback of $\xi:\Omega_{X/\Bbbk}\to\mathcal{O}_{X}$
along $x:\mathrm{Spec}(\Bbbk)\hookrightarrow X$ is zero.

######  Lemma 3.11.

Consider the following diagram in $\mathrm{Sch}/\Bbbk$ such that $i:Z\to Y$ is
an immersion and $j:T\hookrightarrow T[\epsilon]$ is the closed immersion via
$\epsilon=0$

${T}$${T[\epsilon]}$${Z}$${Y.}$$\scriptstyle{j}$$\scriptstyle{g}$$\scriptstyle{\rho}$$\scriptstyle{i}$
(63)

Then $\rho:T[\epsilon]\to Y$ factors through $i:Z\to Y$ if and only if the
morphism $g^{*}\mathcal{C}_{Z/Y}\to\epsilon\mathcal{O}_{T}$ in Lemma 01R3 is
zero, where $\epsilon\mathcal{O}_{T}\cong\mathcal{C}_{T/T[\epsilon]}$.

###### Proof.

Assume without loss of generality that $T,Z,Y$ are affine schemes, and
$i:Z\hookrightarrow Y$ is a closed immersion. Let $T=\mathrm{Spec}(A)$,
$Y=\mathrm{Spec}(R)$ and $Z=\mathrm{Spec}(R/I)$. Let $\varphi:R/I\to A$
correspond to $g:T\to Z$, and let $\psi:R\to A[\epsilon]$ corresponds to
$\rho:T[\epsilon]\to Y$.

Since $\rho\circ j=i\circ g$, the right square in the following diagram
commutes

${0}$${I}$${R}$${R/I}$${0}$${0}$${\epsilon
A}$${A[\epsilon]}$${A}$${0}$$\scriptstyle{\psi|_{I}}$$\scriptstyle{\psi}$$\scriptstyle{\varphi}$
(64)

and thus $\psi|_{I}:I\to\epsilon A$ exists so that the left square commutes.

The morphism $g^{*}\mathcal{C}_{Z/Y}\to\epsilon\mathcal{O}_{T}$ corresponds to
the following

$\eta:I/I^{2}\otimes_{R/I,\varphi}A\to\epsilon A,\quad\overline{h}\otimes
1\mapsto\psi|_{I}(h).$ (65)

Then we have $\eta=0$ if and only if $\psi(I)=0$.

The morphism $\rho:T[\epsilon]\to Y$ factors as $T[\epsilon]\to Z\to Y$ if and
only if the ring map $\psi:R\to A[\epsilon]$ factors as $R\to R/I\to
A[\epsilon]$, which is equivalent to $\psi(I)=0$, and thus equivalent to
$\eta=0$. This completes the proof. ∎

######  Lemma 3.12.

Let $\sigma:G\times X\to X$ be an action. Let $t:T\to X$ be a $T$-valued
point. Then we have the following commutative diagram of
$\Gamma(T,\mathcal{O}_{T})$-modules

${\\{\rho:\rho\textrm{ is a }T\textrm{-valued infinitesimal
transformation}\\}}$${\Gamma(T,\mathfrak{g}\otimes_{\Bbbk}\mathcal{O}_{T})}$${\Big{\\{}\begin{matrix}\rho:\;\rho\textrm{
is a }T\textrm{-valued infinitesimal transformation}\\\ \textrm{and
}\rho\textrm{ stabilises
}t\end{matrix}\Big{\\}}}$${\Gamma\big{(}T,\mathrm{Stab}_{\mathfrak{g}}(t)\big{)}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\subseteq}$$\scriptstyle{\subseteq}$
(66)

where both horizontal maps are bijections and the top map sends
$\rho:T[\epsilon]\to G$ to the morphism of conormal sheaves
$C_{\rho}:\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{T}\to\epsilon\mathcal{O}_{T}$
by applying Lemma 01R3 to the diagram

${T}$${T[\epsilon]}$${T}$${G\times
T.}$$\scriptstyle{\rho}$$\scriptstyle{(e,1_{T})}$ (67)

Moreover, the maps are natural for $T\in\mathrm{Sch}/X$.

###### Proof.

It is easy to see the maps are $\Gamma(T,\mathcal{O}_{T})$-linear and are
natural for $T\in\mathrm{Sch}/X$.

We construct the inverse of the top map. Let
$\eta\in\Gamma(T,\mathfrak{g}\otimes_{\Bbbk}\mathcal{O}_{T})$. We can think
$\eta$ as a map of $\Gamma(T,\mathcal{O}_{T})$-modules

$\eta:\mathfrak{g}^{*}\otimes_{\Bbbk}\Gamma(T,\mathcal{O}_{T})\to\Gamma(T,\mathcal{O}_{T}).$
(68)

Consider the ring map
$\rho^{*}:\mathcal{O}(G)\to\Gamma(T,\mathcal{O}_{T})[\epsilon]$ such that for
$h\in\mathcal{O}(G)$

$\rho^{*}(h)=e^{*}(h)\oplus\epsilon\;\eta\big{(}\overline{h-e^{*}(h)}\otimes
1\big{)}\in\Gamma(T,\mathcal{O}_{T})\oplus\epsilon\Gamma(T,\mathcal{O}_{T})\cong\Gamma(T[\epsilon],\mathcal{O}_{T[\epsilon]})$
(69)

where $h-e^{*}(h)\in\mathfrak{m}_{e}:=\ker(e^{*})$ represents
$\overline{h-e^{*}(h)}\in\mathfrak{m}_{e}/\mathfrak{m}_{e}^{2}\cong\mathfrak{g}^{*}$.
The ring map $\rho^{*}$ corresponds to a morphism $\rho:T[\epsilon]\to G$. It
is easy to check that the map $\eta\mapsto\rho$ provides the required inverse.
This establishes the top bijection.

Let $\rho:T[\epsilon]\to G$ be a $T$-valued infinitesimal transformation. The
following diagram commutes

${T}$${T[\epsilon]}$${X}$${G\times
X}$${\mathrm{Spec}(\Bbbk)}$${G}$${X}$${X\times
X}$$\scriptstyle{t}$$\scriptstyle{(\rho,t\circ\mathrm{pr}_{T})}$$\scriptstyle{\gamma}$$\scriptstyle{(\sigma,\mathrm{pr}_{X})}$$\scriptstyle{e}$$\scriptstyle{\delta}$
(70)

where $\gamma:X\to G\times X$ is the pullback of
$e:\mathrm{Spec}(\Bbbk)\hookrightarrow G$ in the bottom left parallelogram,
$\delta:X\to X\times X$ is the diagonal morphism,
$\mathrm{pr}_{T}:T[\epsilon]\to T$ is the projection and
$\mathrm{pr}_{X}:G\times X\to X$ is the projection.

Apply Lemma 01R3 to the top square and the composition of the top square and
bottom left parallelogram to obtain two morphisms between conormal sheaves

$\begin{split}\mathcal{C}_{(\rho,t\circ\mathrm{pr}_{T})}:&\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{T}\to\epsilon\mathcal{O}_{T}\\\
\mathcal{C}_{\rho}:&\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{T}\to\epsilon\mathcal{O}_{T}.\end{split}$
(71)

such that $\mathcal{C}_{\rho}=\mathcal{C}_{(\rho,t\circ\mathrm{pr}_{T})}\circ
t^{*}\mathcal{C}_{\mathrm{pr}_{G}}$, where
$\mathcal{C}_{\mathrm{pr}_{G}}:\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}\to\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}$
is the morphism of conormal sheaves associated to the bottom right
parallelogram. Since the bottom left parallelogram is Cartesian with southwest
arrows flat, by Lemma 01R4, we have
$\mathcal{C}_{\mathrm{pr}_{G}}=1_{\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}}$.
Then $\mathcal{C}_{\rho}=\mathcal{C}_{(\rho,t\circ\mathrm{pr}_{T})}$.

We have that $\rho$ stabilises $t:T\to X$ if and only if the dashed arrow
below exists in the following diagram

${T}$${T[\epsilon]}$${X}$${X\times
X}$$\scriptstyle{f}$$\scriptstyle{\big{(}\sigma\circ(\rho,t\circ\mathrm{pr}_{T}),\;t\circ\mathrm{pr}_{T}\big{)}}$$\scriptstyle{\delta}$
(72)

where the square (72) is the composition of the top and bottom right
parallelogram in diagram (70). Note that
$\phi:\Omega_{X/\Bbbk}\to\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}$ is
the morphism of conormal sheaves associated to the bottom right parallelogram.
By Lemma 3.11, we have that the dashed arrow above exists if and only if
$t^{*}\Omega_{X/\Bbbk}\to\epsilon\mathcal{O}_{T}$ is zero, i.e. the following
composition is zero

${t^{*}\Omega_{X/\Bbbk}}$${\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{T}}$${\epsilon\mathcal{O}_{T}.}$$\scriptstyle{t^{*}\phi}$$\scriptstyle{\mathcal{C}_{(\rho,t\circ\mathrm{pr}_{T})}}$
(73)

Therefore $\rho$ stabilises $t$ if and only if $\mathcal{C}_{\rho}\circ
t^{*}\phi=\mathcal{C}_{(\rho,t\circ\mathrm{pr}_{T})}\circ t^{*}\phi=0$. Recall
that $\rho\mapsto\mathcal{C}_{\rho}$ defines a bijection

$\\{\rho:\rho\textrm{ is a }T\textrm{-valued infinitesimal
transformation}\\}\cong\mathrm{Hom}_{\mathcal{O}_{T}}(\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{T},\epsilon\mathcal{O}_{T}).$
(74)

Then the images of those $\rho$ which stabilise $t$ constitute the kernel of

$\mathrm{Hom}_{\mathcal{O}_{T}}(\mathfrak{g}^{*}\otimes_{\Bbbk}\mathcal{O}_{T},\mathcal{O}_{T})\to\mathrm{Hom}_{\mathcal{O}_{T}}(t^{*}\Omega_{X/\Bbbk},\mathcal{O}_{T})$
(75)

which is
$\mathrm{Hom}_{\mathcal{O}_{T}}(t^{*}\mathrm{coker}(\phi),\mathcal{O}_{T})\cong\Gamma\big{(}T,\mathrm{Stab}_{\mathfrak{g}}(t)\big{)}$.
This establishes the bottom bijection in the lemma. ∎

###  3.2 Some conditions for the linearisation

From now on, all schemes are of finite type over $\Bbbk$. Let $X$ be a
projective scheme and let $L$ be an ample line bundle on $X$. Let $\hat{U}$
act on $X$ linearly with respect to $L$. In [7], conditions called _Upstairs
Unipotent Stabiliser_ ([7] Assumption 4.40) and _Weak Upstairs Unipotent
Stabiliser_ ([7] Assumption 4.42) were considered for the linearisation
$\hat{U}\curvearrowright(X,L)$, when $X$ is reduced. We will generalise these
conditions for potentially non-reduced $X$. Firstly, a non-emptiness condition
is required, which always holds when $X$ is reduced but can fail in non-
reduced cases.

####  3.2.1 A non-emptiness condition

We will define an open subscheme $X^{0}_{\min}\subseteq X$ and a closed
subscheme $Z_{\min}\hookrightarrow X$. We need a condition to ensure that
$X^{0}_{\min}\neq\emptyset$ and $Z_{\min}\neq\emptyset$.

We first consider when $\hat{U}$ acts on projective spaces. Let $V$ be a
finite dimensional representation of $\hat{U}$. For any $w\in\mathbb{Z}$
denote

$V_{\lambda=w}:=\\{v\in V:\lambda(t).v=t^{w}v,\;\textrm{for all closed point
}t\in\mathbb{G}_{m}\\}.$ (76)

We call $w\in\mathbb{Z}$ a _weight_ if $V_{\lambda=w}\neq 0$, and we call
$V_{\lambda=w}\subseteq V$ the weight subspace of the weight $w$ with respect
to $\lambda:\mathbb{G}_{m}\to\hat{U}$. Let $V_{\lambda=\min}\subseteq V$
denote the weight subspace whose weight is the minimum. If $V\neq 0$, then
$V_{\min}\neq 0$. Let $w_{\min}$ denote the minimal weight. For $v\in V$,
decompose according to $\mathbb{G}_{m}$-weights

$v=v_{\lambda=\min}+v_{\lambda>\min},\quad v_{\lambda=\min}\in
V_{\lambda=\min},\;v_{\lambda>\min}\in\bigoplus_{w>w_{\min}}V_{\lambda=w}.$
(77)

Define locally closed subschemes
$\mathbb{P}(V)^{0}_{\min}\subseteq\mathbb{P}(V)$ and
$Z_{\min}\big{(}\mathbb{P}(V)\big{)}\hookrightarrow\mathbb{P}(V)$ whose sets
of closed points are

$\begin{split}\mathbb{P}(V)^{0}_{\min}\overset{\mathrm{cl}}{=\\!=}&\big{\\{}[v]\in\mathbb{P}(V):v_{\lambda=\min}\neq
0\big{\\}}\\\
Z_{\min}\big{(}\mathbb{P}(V)\big{)}\overset{\mathrm{cl}}{=\\!=}&\big{\\{}[v]\in\mathbb{P}(V):v_{\lambda>\min}=0\big{\\}}\end{split}$
(78)

where $\overset{\mathrm{cl}}{=\\!=}$ means they have the same closed points.
It is easy to see that $\mathbb{P}(V)^{0}_{\min}$ is an open subscheme and
$Z_{\min}\big{(}\mathbb{P}(V)\big{)}$ is a closed subscheme, and
$Z_{\min}\big{(}\mathbb{P}(V)\big{)}\hookrightarrow\mathbb{P}(V)^{0}_{\min}$.

The non-emptiness condition for projective spaces is vacuous, since if
$\mathbb{P}(V)\neq\emptyset$, then $\mathbb{P}(V)^{0}_{\min}\neq\emptyset$ and
$Z_{\min}\big{(}\mathbb{P}(V)\big{)}\neq\emptyset$. It is vacuous if $X$ is
reduced.

Then we consider the linearisation $\hat{U}\curvearrowright(X,L)$. There
exists $m\in\mathbb{N}_{+}$ such that $H^{0}(X,L^{m})$ generates
$\bigoplus_{r\in\mathbb{N}}H^{0}(X,L^{mr})$. We can define
$\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}^{0}_{\min}$ and
$Z_{\min}\big{(}\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}\big{)}$ as above.

######  Lemma 3.13.

For the linearisation $\hat{U}\curvearrowright(X,L)$, if $m\in\mathbb{N}_{+}$
is such that $H^{0}(X,L^{m})$ generates
$\bigoplus_{r\in\mathbb{N}}H^{0}(X,L^{mr})$ and

$X\cap\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}^{0}_{\min}\neq\emptyset$ (79)

where $X\hookrightarrow\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}$ is a closed
immersion, then for all $n\in\mathbb{N}_{+}$ we have

$X\cap\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}^{0}_{\min}=X\cap\mathbb{P}\big{(}H^{0}(X,L^{mn})^{*}\big{)}^{0}_{\min}.$
(80)

###### Proof.

Let $m\in\mathbb{N}_{+}$ be as in the lemma. The linear system
$H^{0}(X,L^{mn})$ induces a closed immersion
$X\hookrightarrow\mathbb{P}\big{(}H^{0}(X,L^{mn})^{*}\big{)}$ for all
$n\in\mathbb{N}_{+}$.

The open subscheme
$X\cap\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}^{0}_{\min}$ is the non-
vanishing locus of $H^{0}(X,L^{m})_{\lambda=\max}$, where
$H^{0}(X,L^{m})_{\lambda=\max}$ is the weight subspace of $H^{0}(X,L^{m})$
with respect to $\lambda$ whose weight is the maximum. Similarly, the open
subscheme $X\cap\mathbb{P}\big{(}H^{0}(X,L^{mn})^{*}\big{)}^{0}_{\min}$ is the
non-vanishing locus of $H^{0}(X,L^{mn})_{\lambda=\max}$.

Let $w_{\max}$ denote the maximal weight of $\lambda$ on $H^{0}(X,L^{m})$. The
condition that
$X\cap\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}^{0}_{\min}\neq\emptyset$
implies that there exists $f\in H^{0}(X,L^{m})_{\lambda=\max}$, which is not
nilpotent in $\bigoplus_{r\in\mathbb{N}_{+}}H^{0}(X,L^{r})$. Therefore
$f^{n}\in H^{0}(X,L^{mn})\setminus\\{0\\}$. Since $H^{0}(X,L^{m})$ generates
$\bigoplus_{r\in\mathbb{N}_{+}}H^{0}(X,L^{mr})$, we have that the maximal
weight on $H^{0}(X,L^{mn})$ is the weight of $f^{n}$, i.e. $nw_{\max}$. We
have the following surjective map

$\mathrm{Sym}^{n}H^{0}(X,L^{m})_{\lambda=\max}\to
H^{0}(X,L^{mn})_{\lambda=\max}.$ (81)

Therefore the non-vanishing locus of $H^{0}(X,L^{mn})_{\lambda=\max}$ and the
non-vanishing locus of $H^{0}(X,L^{m})_{\lambda=\max}$ coincide. This proves
the lemma. ∎

######  Condition 3.14.

For the linearisation $\hat{U}\curvearrowright(X,L)$, assume that

$X\cap\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}^{0}_{\min}\neq\emptyset$ (82)

for sufficiently divisible $m\in\mathbb{N}_{+}$. Denote

$X^{0}_{\min}:=X\cap\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}.$ (83)

######  Remark 3.15.

Lemma 3.13 implies that $X^{0}_{\min}$ does not depend on $m\in\mathbb{N}_{+}$
when $m$ is sufficiently divisible and when Condition 3.14 holds.

The opposite of Condition 3.14 is the condition that
$H^{0}(X,L^{m})_{\lambda=\max}$ is nilpotent for all sufficiently divisible
$m\in\mathbb{N}_{+}$. When $X$ is non-empty and reduced, then Condition 3.14
always holds.

With Condition 3.14, we also have that $X^{0}_{\min}$ is the non-vanishing
locus of $\bigoplus_{r\in\mathbb{N}_{+}}H^{0}(X,L^{r})_{\lambda=\max}$. Define
the _fixed point subscheme_ $Z_{\min}\hookrightarrow X^{0}_{\min}$

$Z_{\min}:=\big{(}X^{0}_{\min}\big{)}^{\lambda}.$ (84)

Then
$Z_{\min}=\mathrm{Proj}\Big{(}\bigoplus_{r\in\mathbb{N}_{+}}H^{0}(X,L^{r})_{\lambda=\max}\Big{)}$.

####  3.2.2 Upstairs Unipotent Stabiliser conditions

Let $\phi:\Omega_{X/\Bbbk}\to\mathfrak{u}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}$
be the infinitesimal action of $\mathfrak{u}$ on $X$ (Definition 3.2), and let
$\mathrm{Fit}_{k}(\phi)\subseteq\mathcal{O}_{X}$ denote the $k$th Fitting
ideal of $\mathrm{coker}(\phi)$. The following conditions generalise [6]
Assumption 4.40 and Assumption 4.42.

######  Condition 3.16 (Upstairs Unipotent Stabiliser).

For the linearisation $\hat{U}\curvearrowright(X,L)$ satisfying Condition
3.14, we say the _upstairs unipotent stabiliser condition_ (Condition UU)
holds if $\mathrm{coker}(\phi)$ is locally free of constant rank on
$X^{0}_{\min}$.

######  Remark 3.17.

By Lemma 3.6, Condition UU implies that $\dim\mathrm{Stab}_{U}(x)$ is constant
for $\Bbbk$-points $x\in X^{0}_{\min}$. When $X^{0}_{\min}$ is reduced, the
converse is true.

######  Condition 3.18 (Weak Upstairs Unipotent Stabiliser).

For the linearisation $\hat{U}\curvearrowright(X,L)$ satisfying Condition
3.14, we say the _weak upstairs unipotent stabiliser condition_ (Condition
WUU) holds if $\mathrm{coker}(\phi)$ is locally free of rank $k$ on an open
subscheme which meets $Z_{\min}$, where
$k:=\min\\{i\in\mathbb{N}:\mathrm{Fit}_{i}(\phi)\neq 0\\}$.

###  3.3 Theorem with Condition UU

When the linearisation $\hat{U}\curvearrowright(X,L)$ satisfies Condition UU,
the open subscheme $X^{0}_{\min}$ admits a geometric quotient by $U$, and the
open subscheme $X^{0}_{\min}\setminus UZ_{\min}$ admits a projective geometric
quotient by $\hat{U}$. We will prove these statements.

Let $r:=\dim U$. Without loss of generality assume $r>0$. Let
$\phi:\Omega_{X/\Bbbk}\to\mathfrak{u}^{*}\otimes_{\Bbbk}\mathcal{O}_{X}$ be
the infinitesimal action of $\mathfrak{u}$. Condition UU implies that
$X^{0}_{\min}\neq\emptyset$.

####  3.3.1 Reduction to affine case

Condition UU assumes that $\mathrm{coker}(\phi)$ is locally free of rank $k$
on $X^{0}_{\min}$ for some integer $0\leq k\leq r$. The quotient
$\mathfrak{u}^{*}\otimes_{\Bbbk}\mathcal{O}_{X^{0}_{\min}}\to\mathrm{coker}(\phi)|_{X^{0}_{\min}}$
corresponds to a morphism

$\Theta:X^{0}_{\min}\to\mathrm{Grass}(\mathfrak{u}^{*},k)$ (85)

where $\mathrm{Grass}(\mathfrak{u}^{*},k)$ is the Grassmannian of
$k$-dimensional quotients of $\mathfrak{u}^{*}$. Let
$\iota:Z_{\min}\hookrightarrow X^{0}_{\min}$ be the closed immersion.

There exists a finite family of $k$-dimensional vector subspaces
$\big{\\{}\mathfrak{s}_{i}^{*}\subseteq\mathfrak{u}^{*}\\}_{i=1}^{s}$ such
that

$\mathrm{Grass}(\mathfrak{u}^{*},k)=\bigcup_{i=1}^{s}\mathrm{Grass}(\mathfrak{u}^{*},k)_{\mathfrak{s}_{i}^{*}}$
(86)

where
$\mathrm{Grass}(\mathfrak{u}^{*},k)_{\mathfrak{s}_{i}^{*}}\subseteq\mathrm{Grass}(\mathfrak{u}^{*},k)$
is the open subscheme whose closed points are quotients $\mathfrak{u}^{*}\to
Q$ such that the composition $\mathfrak{s}_{i}^{*}\subseteq\mathfrak{u}^{*}\to
Q$ is an isomorphism.

Open subsets $Z_{\min,f}\subseteq Z_{\min}$ for $m\in\mathbb{N}_{+}$ and $f\in
H^{0}(X,L^{m})_{\lambda=\max}$ form a basis of the Zariski topology of
$Z_{\min}$. We can choose a sufficiently divisible $m\in\mathbb{N}_{+}$ and
$f_{1},\cdots,f_{t}\in H^{0}(X,L^{m})_{\lambda=\max}$ such that
$\\{Z_{\min,f_{i}}\\}_{i=1}^{t}$ is a covering of $Z_{\min}$, and the covering
is subordinate to the covering of $\mathrm{Grass}(\mathfrak{u}^{*},k)$, in the
sense that for any $1\leq i\leq t$, there exists $1\leq j\leq s$ such that

$Z_{\min,f_{i}}\subseteq\Theta^{-1}\Big{(}\mathrm{Grass}(\mathfrak{u}^{*},k)_{\mathfrak{s}_{j}^{*}}\Big{)}.$
(87)

The morphism $\Theta$ is $\mathbb{G}_{m}$-invariant and
$\mathbb{G}_{m}$-equivariant, so

$\begin{split}&Z_{\min,f_{i}}\subseteq\Theta^{-1}\Big{(}\mathrm{Grass}(\mathfrak{u}^{*},k)_{\mathfrak{s}_{j}^{*}}\Big{)}\\\
\textrm{if and only if
}\;&X_{f_{i}}\subseteq\Theta^{-1}\Big{(}\mathrm{Grass}(\mathfrak{u}^{*},k)_{\mathfrak{s}_{j}^{*}}\Big{)}.\end{split}$
(88)

We have achieved the following:

  * •

There exists $m\in\mathbb{N}_{+}$ and $f_{1},\cdots,f_{t}\in
H^{0}(X,L^{m})_{\lambda=\max}$ such that $X^{0}_{\min}$ is covered by
$\\{X_{f_{i}}\\}_{i=1}^{t}$;

  * •

For each $X_{f_{i}}$, there exists a linear subspace
$\mathfrak{s}_{i}^{*}\subseteq\mathfrak{u}^{*}$ such that
$\mathfrak{s}_{i}^{*}\otimes_{\Bbbk}\mathcal{O}_{X_{f_{i}}}\cong\mathrm{coker}(\phi)|_{X_{f_{i}}}$
as follows

${\mathfrak{s}_{i}^{*}\otimes_{\Bbbk}\mathcal{O}_{X_{f_{i}}}}$${\Omega_{X_{f_{i}}/\Bbbk}}$${\mathfrak{u}^{*}\otimes_{\Bbbk}\mathcal{O}_{X_{f_{i}}}}$${\mathrm{coker}(\phi)|_{X_{f_{i}}}}$${0.}$$\scriptstyle{\subseteq}$$\scriptstyle{\cong}$$\scriptstyle{\phi|_{X_{f_{i}}}}$
(89)

####  3.3.2 Affine theory with Condition UU

We work on each of $X_{f_{i}}$ and construct a universal geometric quotient by
$U$.

Let $A$ be a finitely generated commutative $\Bbbk$-algebra. Let $\hat{U}$ act
on $\mathrm{Spec}(A)$. Assume the following conditions for
$\hat{U}\curvearrowright\mathrm{Spec}(A)$:

  * (1)

If $A_{\lambda=w}\subseteq A$ denotes the weight subspace whose
$\lambda$-weight is $w\in\mathbb{Z}$ for the $\mathbb{G}_{m}$-representation
on $A$, then $A_{\lambda=w}=0$ for $w>0$;

  * (2)

There exists a linear subspace $\mathfrak{s}^{*}\subseteq\mathfrak{u}^{*}$
such that $\mathfrak{s}^{*}\otimes_{\Bbbk}A\cong\mathrm{coker}(\phi_{A})$ in
the following diagram

${\mathfrak{s}^{*}\otimes_{\Bbbk}A}$${\Omega_{A/\Bbbk}}$${\mathfrak{u}^{*}\otimes_{\Bbbk}A}$${\mathrm{coker}(\phi_{A})}$${0.}$$\scriptstyle{\subseteq}$$\scriptstyle{\cong}$$\scriptstyle{\phi_{A}}$
(90)

The action $\hat{U}\curvearrowright X_{f_{i}}$ satisfies these conditions.

######  Remark 3.19.

If $\rho:A\to\Bbbk[t,t^{-1}]\otimes_{\Bbbk}A$ denotes the co-action map of
$\mathbb{G}_{m}\curvearrowright\mathrm{Spec}(A)$, then $A_{\lambda=w}=\\{f\in
A:\rho(f)=t^{-w}\otimes f\\}$. Condition $(1)$ above is equivalent to that
$\rho$ factors through $\Bbbk[t]\otimes_{\Bbbk}A$

${A}$${\Bbbk[t,t^{-1}]\otimes_{\Bbbk}A}$${\Bbbk[t]\otimes_{\Bbbk}A.}$$\scriptstyle{\rho}$$\scriptstyle{\subseteq}$
(91)

Geometrically, Condition $(1)$ is equivalent to that the action morphism
$\mathbb{G}_{m}\times\mathrm{Spec}(A)\to\mathrm{Spec}(A)$ extends to a
morphism $\mathbb{A}^{1}\times\mathrm{Spec}(A)\to\mathrm{Spec}(A)$.

Let $\mathfrak{u}\to\mathfrak{s}$ denote the surjective linear map dual to
$\mathfrak{s}^{*}\subseteq\mathfrak{u}^{*}$. Let
$\mathfrak{u}^{\prime}:=\ker(\mathfrak{u}\to\mathfrak{s})$. Let
$\sigma^{*}:A\to\mathcal{O}(U)\otimes_{\Bbbk}A$ denote the co-action map
associated to $U\curvearrowright\mathrm{Spec}(A)$. Define sub-rings invariant
under $\mathfrak{u}^{\prime}$ and $\mathfrak{u}$

$\begin{split}A^{\mathfrak{u}^{\prime}}:=&\\{f\in A:\xi.f=0\textrm{ for all
}\xi\in\mathfrak{u}^{\prime}\\}\\\ A^{\mathfrak{u}}:=&\\{f\in
A:\xi.f=0\textrm{ for all }\xi\in\mathfrak{u}\\}\end{split}$ (92)

where $\xi:A\to A$ is the $\Bbbk$-derivation defined as follows

${A}$${\mathcal{O}(U)\otimes_{\Bbbk}A}$${\mathfrak{u}^{*}\otimes_{\Bbbk}A}$${A}$$\scriptstyle{\xi}$$\scriptstyle{\sigma^{*}}$$\scriptstyle{\exp^{*}\otimes
1_{A}}$$\scriptstyle{-\xi\otimes 1_{A}}$ (93)

for $\exp^{*}:\mathcal{O}(U)\to\mathfrak{u}^{*}$ sending $f$ to
$\overline{f-f(e)}\in\mathfrak{m}_{e}/\mathfrak{m}_{e}^{2}\cong\mathfrak{u}^{*}$.
It is easy to check $A^{\mathfrak{u}}\subseteq A^{\mathfrak{u}^{\prime}}$ are
sub-rings.

######  Lemma 3.20.

We have $A^{\mathfrak{u}}=A^{\mathfrak{u}^{\prime}}$.

###### Proof.

Let $\\{\xi_{1},\cdots,\xi_{r}\\}$ be a basis of $\mathfrak{u}$ such that
$\xi_{1},\cdots,\xi_{r-k}\in\mathfrak{u}^{\prime}$. Let
$\\{u_{1},\cdots,u_{r}\\}$ be the basis of $\mathfrak{u}^{*}$ dual to
$\\{\xi_{1},\cdots,\xi_{r}\\}$. Then $\\{\xi_{r-k+1},\cdots,\xi_{r}\\}$ is a
basis of $\mathfrak{s}^{*}$. The infinitesimal action
$\phi_{A}:\Omega_{A/\Bbbk}\to\mathfrak{u}^{*}\otimes_{\Bbbk}A$ is determined
by the equation

$\phi_{A}(\mathrm{d}f)=\sum_{i=1}^{r}u_{i}\otimes\xi_{i}.f,\quad f\in A.$ (94)

For $\xi\in\mathfrak{u}$ and $f\in A$ we have

$\langle\phi_{A}(\mathrm{d}f),\;\xi\rangle=\sum_{i=1}^{r}\langle
u_{i},\xi\rangle\otimes\xi_{i}.f=\xi.f$ (95)

and therefore we have

$\begin{split}A^{\mathfrak{u}^{\prime}}=&\\{f\in
A:\phi_{A}(\mathrm{d}f)\subseteq\mathfrak{s}^{*}\otimes_{\Bbbk}A\\}\\\
A^{\mathfrak{u}}=&\\{f\in A:\phi_{A}(\mathrm{d}f)=0\\}.\end{split}$ (96)

By diagram (90), we have
$\mathrm{im}(\phi_{A})\cap(\mathfrak{s}^{*}\otimes_{\Bbbk}A)=0$. The lemma is
proved. ∎

We first construct the quotient of $\mathrm{Spec}(A)$ by the subgroup
$U^{\prime}:=\mathrm{exp}(\mathfrak{u}^{\prime})\subseteq U$.

######  Lemma 3.21.

There exists $\xi_{1},\cdots,\xi_{r-k}\in\mathfrak{u}^{\prime}$ and
$x_{1},\cdots,x_{r-k}\in A$ such that

  * •

$\xi_{i}.x_{j}=\delta_{ij}$ for $1\leq i,j\leq r-k$;

  * •

$\xi.x_{j}\in A_{\lambda=0}$ for $\xi\in\mathfrak{u}$ and $1\leq j\leq r-k$.

###### Proof.

Let $w\in\mathbb{N}_{+}$ be the unique $\lambda$-weight on $\mathfrak{u}$. Let
$\\{\xi_{1},\cdots,\xi_{r}\\}$ be a basis of $\mathfrak{u}$ such that
$\xi_{1},\cdots,\xi_{r-k}\in\mathfrak{u}^{\prime}$. Let
$\\{u_{1},\cdots,u_{r}\\}$ be the basis of $\mathfrak{u}^{*}$ dual to
$\\{\xi_{1},\cdots,\xi_{r}\\}$.

By diagram (90), the following composition is surjective

${\Omega_{A/\Bbbk}}$${\mathfrak{u}^{*}\otimes_{\Bbbk}A}$${(\mathfrak{u}^{\prime})^{*}\otimes_{\Bbbk}A.}$$\scriptstyle{\phi_{A}}$
(97)

Then for $1\leq i\leq r-k$, there exists $\omega_{i}\in\Omega_{A/\Bbbk}$ such
that

$\phi_{A}(\omega_{i})\in u_{i}\otimes 1+\mathfrak{s}^{*}\otimes_{\Bbbk}A.$
(98)

Write $\omega_{i}=\sum_{n,\mu,\nu}a^{(i)}_{n,\mu}\;\mathrm{d}b^{(i)}_{n,\nu}$
for $a^{(i)}_{n,\mu}\in A_{\lambda=\mu},b^{(i)}_{n,\nu}\in A_{\lambda=\nu}$.
Then we have

$\begin{split}\phi_{A}(\omega_{i})=&\sum_{j=1}^{r}\sum_{n,\mu,\nu}u_{j}\otimes
a^{(i)}_{n,\mu}\;\xi_{j}.b^{(i)}_{n,\nu}\\\
=&\sum_{j=1}^{r}\sum_{n}u_{j}\otimes
a^{(i)}_{n,0}\;\xi_{j}.b^{(i)}_{n,-w}+\sum_{j=1}^{r}\sum_{\begin{subarray}{c}n,\mu,\nu\\\
\mu<0\\\ \textrm{or }\nu<-w\end{subarray}}u_{j}\otimes
a^{(i)}_{n,\mu}\;\xi_{j}.b^{(i)}_{n,\nu}.\end{split}$ (99)

Let $x_{i}:=\sum_{n}a^{(i)}_{n,0}b^{(i)}_{n,-w}\in A_{\lambda=-w}$ for $1\leq
i\leq r-k$. Since $x_{i}\in A_{\lambda=-w}$ and any
$\xi\in\mathfrak{u}\setminus\\{0\\}$ has weight $w\in\mathbb{N}_{+}$, we have
$\xi.x_{i}\in A_{\lambda=0}$. This checks the second condition in the lemma.

Let $p:A\to A_{\lambda=0}$ be the projection onto the weight subspace of
weight 0, and let $q:\mathfrak{u}^{*}\to(\mathfrak{u}^{\prime})^{*}$ be the
surjective linear map. Then in
$(\mathfrak{u}^{\prime})^{*}\otimes_{\Bbbk}A_{\lambda=0}$

$\begin{split}u_{i}\otimes 1=&(q\otimes p)\circ\phi_{A}(\omega_{i})\\\
=&(q\otimes p)\Big{(}\sum_{j=1}^{r}\sum_{n}u_{j}\otimes
a^{(i)}_{n,0}\;\xi_{j}.b^{(i)}_{n,-w}+\sum_{j=1}^{r}\sum_{\begin{subarray}{c}n,\mu,\nu\\\
\mu<0\\\ \textrm{or }\nu<-w\end{subarray}}u_{j}\otimes
a^{(i)}_{n,\mu}\;\xi_{j}.b^{(i)}_{n,\nu}\Big{)}\\\
=&\sum_{j=1}^{r-k}u_{j}\otimes\xi_{j}.\Big{(}\sum_{n}a^{(i)}_{n,0}b^{(i)}_{n,-w}\Big{)}\\\
=&\sum_{j=1}^{r-k}u_{j}\otimes\xi_{j}.x_{i}\end{split}$ (100)

that is $\xi_{i}.x_{j}=\delta_{ij}$ for $1\leq i,j\leq r-k$. This checks the
first condition in the lemma. ∎

######  Proposition 3.22.

Let $\mathrm{Spec}(A)$ be an algebraic affine scheme over $\Bbbk$. Let
$\hat{U}$ act on $\mathrm{Spec}(A)$. Assume the action
$\hat{U}\curvearrowright\mathrm{Spec}(A)$ satisfies:

  * (1)

$A_{\lambda=w}=0$ if $w>0$;

  * (2)

There exists a subspace $\mathfrak{s}^{*}\subseteq\mathfrak{u}^{*}$ such that
$\mathfrak{s}^{*}\otimes_{\Bbbk}A\cong\mathrm{coker}(\phi_{A})$ in the diagram

${\mathfrak{s}^{*}\otimes_{\Bbbk}A}$${\Omega_{A/\Bbbk}}$${\mathfrak{u}^{*}\otimes_{\Bbbk}A}$${\mathrm{coker}(\phi_{A})}$${0}$$\scriptstyle{\subseteq}$$\scriptstyle{\cong}$$\scriptstyle{\phi_{A}}$
(101)

where $\phi_{A}$ is the infinitesimal action of $\mathfrak{u}$ on
$\mathrm{Spec}(A)$.

Then:

  * •

$A$ is a polynomial ring over $A^{\mathfrak{u}}$;

  * •

The affine morphism $\mathrm{Spec}(A)\to\mathrm{Spec}(A^{\mathfrak{u}})$ is a
universal geometric quotient by $U$.

###### Proof.

Let $\mathfrak{u}^{\prime}:=\ker(\mathfrak{u}\to\mathfrak{s})$, where
$\mathfrak{u}\to\mathfrak{s}$ is dual to
$\mathfrak{s}^{*}\subseteq\mathfrak{u}^{*}$. We can apply [5] Theorem 3.10 for
the action of $\mathfrak{u}^{\prime}$ on $A$. Lemma 3.21 shows that Condition
$(3)$ in [5] Theorem 3.10 is satisfied and then we have Condition $(6)$ in
that theorem that
$\mathrm{Spec}(A)\to\mathrm{Spec}(A^{\mathfrak{u}^{\prime}})$ is a trivial
geometric quotient by $U^{\prime}=\exp(\mathfrak{u}^{\prime})$. Actually
according to the proof of [5] Theorem 3.10, we have that
$x_{1},\cdots,x_{r-k}\in A$ are algebraically independent over
$A^{\mathfrak{u}^{\prime}}$ and
$A=A^{\mathfrak{u}^{\prime}}[x_{1},\cdots,x_{r-k}]$. Since
$A^{\mathfrak{u}}=A^{\mathfrak{u}^{\prime}}$ by Lemma 3.20, we have

$A=A^{\mathfrak{u}}[x_{1},\cdots,x_{r-k}].$ (102)

This proves that $A$ is a polynomial ring over $A^{\mathfrak{u}}$. The
statement that $\mathrm{Spec}(A)\to\mathrm{Spec}(A^{\mathfrak{u}})$ is a
universal geometric quotient follows immediately. ∎

######  Corollary 3.23.

Let $\hat{U}\curvearrowright\mathrm{Spec}(A)$ and
$\mathfrak{s}^{*}\subseteq\mathfrak{u}^{*}$ be as in Proposition 3.22. Let
$X:=\mathrm{Spec}(A)$ and $Y:=\mathrm{Spec}(A^{\mathfrak{u}})$. Then $Y$
represents the quotient presheaf $X/_{p}U$

$X/_{p}U:(\mathrm{Sch}/\Bbbk)^{\mathrm{op}}\to\mathrm{Set},\quad
T\mapsto\frac{\mathrm{Hom}_{\mathrm{Sch}/\Bbbk}(T,X)}{\mathrm{Hom}_{\mathrm{Sch}/\Bbbk}(T,U)}.$
(103)

Moreover for any morphism $Y^{\prime}\to Y$ in $\mathrm{Sch}/\Bbbk$, we have
that $Y^{\prime}$ represents the quotient sheaf $(X^{\prime}/_{p}U)^{\sharp}$
on $(\mathrm{Sch}/\Bbbk)_{\acute{e}tale}$ (See Definition 02VG for quotient
sheaves), where $X^{\prime}:=X\times_{Y}Y^{\prime}$, and
$(X^{\prime}/_{p}U)^{\sharp}$ is the sheafification of $X^{\prime}/_{p}U$.

###### Proof.

For a scheme, we think it as a sheaf on $(\mathrm{Sch}/\Bbbk)_{\acute{e}tale}$
via its functor of points. Let $\pi:X\to Y$ be the geometric quotient in
Proposition 3.22. Since $\pi:X\to Y$ is $U$-invariant, the map
$\mathrm{Hom}_{\mathrm{Sch}/\Bbbk}(T,X)\to\mathrm{Hom}_{\mathrm{Sch}/\Bbbk}(T,Y)$
factors through $(X/_{p}U)(T)$ for any $T\in\mathrm{Sch}/\Bbbk$. This gives a
morphism $\eta:X/_{p}U\to Y$ of presheaves.

By Lemma 3.21 and Proposition 3.22, we can choose a basis
$\xi_{1},\cdots,\xi_{r}$ of $\mathfrak{u}$ and $x_{1},\cdots,x_{r-k}\in A$
such that:

  * •

$A=A^{\mathfrak{u}}[x_{1},\cdots,x_{r-k}]$ and $x_{1},\cdots,x_{r-k}$ are
algebraically independent over $A^{\mathfrak{u}}$;

  * •

$\xi_{i}.x_{j}=\delta_{ij}$ if $1\leq i,j\leq r-k$.

Let $u_{1},\cdots,u_{r}\in\mathfrak{u}^{*}$ be the basis dual to
$\xi_{1},\cdots,\xi_{r}$. Lift $u_{i}\in\mathfrak{u}^{*}$ to
$u_{i}\in\mathfrak{m}_{e}\subseteq\mathcal{O}(U)$. We identify
$\mathcal{O}(U)\cong\Bbbk[u_{1},\cdots,u_{r}]$. Then for $1\leq i\leq r-k$

$\sigma^{*}:A\to\mathcal{O}(U)\otimes_{\Bbbk}A,\quad x_{i}\mapsto 1\otimes
x_{i}+u_{i}\otimes
1+\mathrm{err}_{i},\quad\mathrm{err}_{i}\in(u_{r-k+1}\otimes
1,\cdots,u_{r}\otimes 1).$ (104)

We first show that $\eta:X/_{p}U\to Y$ is injective. Let
$T\in\mathrm{Sch}/\Bbbk$. Let $g_{i}:T\to X$ for $i=1,2$ be two morphisms such
that $\eta_{T}([g_{1}])=\eta_{T}([g_{2}])=h$, i.e. $h=\pi\circ g_{1}=\pi\circ
g_{2}$. We identify $\mathcal{O}(U)\cong\Bbbk[u_{1},\cdots,u_{r}]$ and define
a ring map

$l^{*}:\Bbbk[u_{1},\cdots,u_{r}]\to\Gamma(T,\mathcal{O}_{T}),\quad
u_{i}\mapsto\begin{cases}g_{2}^{*}(x_{i})-g_{1}^{*}(x_{i}),&\quad 1\leq i\leq
r-k\\\ 0,&\quad r-k<i\leq r.\end{cases}$ (105)

Then the following diagram commutes

${\Gamma(T,\mathcal{O}_{T})}$${A}$${\Gamma(T,\mathcal{O}_{T})\otimes_{\Bbbk}\Gamma(T,\mathcal{O}_{T})}$${\mathcal{O}(U)\otimes_{\Bbbk}A}$${A}$$\scriptstyle{g_{2}^{*}}$$\scriptstyle{\Delta^{*}}$$\scriptstyle{l^{*}\otimes
g_{1}^{*}}$$\scriptstyle{\sigma^{*}}$ (106)

which corresponds to a commutative diagram of schemes

${T}$${X}$${T\times T}$${U\times
X}$${X.}$$\scriptstyle{g_{2}}$$\scriptstyle{\Delta}$$\scriptstyle{l\times
g_{1}}$$\scriptstyle{\sigma}$ (107)

where $l:T\to U$ corresponds to the ring map
$l^{*}:\mathcal{O}(U)\to\Gamma(T,\mathcal{O}_{T})$. Then
$g_{2}=\sigma(l,g_{1})$ for this $l:T\to U$. This implies
$[g_{1}]=[g_{2}]\in(X/_{p}U)(T)$, so $\eta$ is injective.

We then show $\eta$ is surjective as a morphism of presheaves. Let
$T\in\mathrm{Sch}/\Bbbk$. Let $h:T\to Y$ be in
$\mathrm{Hom}_{\mathrm{Sch}/\Bbbk}(T,Y)$. Let
$g^{*}:A\to\Gamma(T,\mathcal{O}_{T})$ be the composition
$A=A^{\mathfrak{u}}[x_{1},\cdots,x_{r-k}]\to
A^{\mathfrak{u}}[x_{1},\cdots,x_{r-k}]/(x_{1},\cdots,x_{r-k})\cong
A^{\mathfrak{u}}\xrightarrow{h^{*}}\Gamma(T,\mathcal{O}_{T})$. Let $g:T\to X$
correspond to $g^{*}:A\to\Gamma(T,\mathcal{O}_{T})$. Then $\eta_{T}([g])=h$.
This shows that $\eta_{T}$ is surjective. Then $\eta:X/_{p}U\cong Y$.

We first prove that if $Y^{\prime}\to Y$ is a morphism between affine schemes,
then $X^{\prime}/_{p}U\cong Y^{\prime}$ as presheaves. Let $Y^{\prime}\to Y$
be a morphism with $Y^{\prime}$ affine with coordinate ring $B$. Then
$X^{\prime}=X\times_{Y}Y^{\prime}$ is affine with coordinate ring
$A\otimes_{A^{\mathfrak{u}}}B=B[x_{1},\cdots,x_{r-k}]$. The same argument as
above with $A^{\mathfrak{u}}$ replaced by $B$ shows that $X^{\prime}/_{p}U\to
Y^{\prime}$ is injective and surjective as presheaves.

Next, we consider a general morphism $Y^{\prime}\to Y$ in
$\mathrm{Sch}/\Bbbk$. Choose a covering $\\{Y^{\prime}_{j}\to
Y^{\prime}\\}_{j\in J}$ in $(\mathrm{Sch}/\Bbbk)_{\acute{e}tale}$ such that
each of $Y^{\prime}_{j}$ is affine. Let $B_{j}$ be the coordinate ring of
$Y^{\prime}_{j}$. The composition $Y^{\prime}_{j}\to Y^{\prime}\to Y$
corresponds to a ring map $A^{\mathfrak{u}}\to B_{j}$. Let
$X^{\prime}_{j}:=X\times_{Y}Y^{\prime}_{j}$. Then $X^{\prime}_{j}$ is affine
with coordinate ring $A\otimes_{A^{\mathfrak{u}}}B_{j}\cong
B_{j}[x_{1},\cdots,x_{r-k}]$.

We construct a morphism of presheaves
$Y^{\prime}\to(X^{\prime}/_{p}U)^{\sharp}$. By the Yoneda lemma, this is
equivalent to an element in $(X^{\prime}/_{p}U)^{\sharp}(Y^{\prime})$. The
sheaf property of $(X^{\prime}/_{p}U)^{\sharp}$ implies that the following is
an equaliser diagram

${(X^{\prime}/_{p}U)^{\sharp}(Y^{\prime})}$${\displaystyle\prod_{j\in
J}(X^{\prime}/_{p}U)^{\sharp}(Y^{\prime}_{j})}$${\displaystyle\prod_{k,l\in
J}(X^{\prime}/_{p}U)^{\sharp}\big{(}Y^{\prime}_{k}\times_{Y^{\prime}}Y^{\prime}_{l}\big{)}.}$
(108)

We have that $X^{\prime}_{j}/_{p}U\cong Y^{\prime}_{j}$ as presheaves. Let
$\zeta_{j}\in(X^{\prime}/_{p}U)^{\sharp}(Y^{\prime}_{j})$ be elements
corresponding to morphisms of presheaves
$\zeta_{j}:Y^{\prime}_{j}\to(X^{\prime}/_{p}U)^{\sharp}$ as follows

${Y^{\prime}_{j}}$${X^{\prime}_{j}/_{p}U}$${X^{\prime}/_{p}U}$${(X^{\prime}/_{p}U)^{\sharp}}$$\scriptstyle{\zeta_{j}}$$\scriptstyle{\cong}$
(109)

where $X^{\prime}_{j}/_{p}U\to X^{\prime}/_{p}U$ is induced by
$X^{\prime}_{j}\to X^{\prime}$, which is the base change of $Y^{\prime}_{j}\to
Y^{\prime}$. We will show the collection $\\{\zeta_{j}\\}_{j\in J}$ is a
descent datum with respect to $\\{Y^{\prime}_{j}\to Y^{\prime}\\}_{j\in J}$.

Choose $k,l\in J$. Denote projections and base changes of covering morphisms

$\begin{split}\mathrm{pr}_{k}:&Y^{\prime}_{k}\times_{Y^{\prime}}Y^{\prime}_{k}\to
Y^{\prime}_{k},\quad\mathrm{pr}_{l}:Y^{\prime}_{k}\times_{Y^{\prime}}Y^{\prime}_{l}\to
Y^{\prime}_{l}\\\ \iota_{k}:&X^{\prime}_{k}\to
X^{\prime},\quad\iota_{l}:X^{\prime}_{l}\to X^{\prime}.\end{split}$ (110)

Since $X^{\prime}_{k}/_{p}U\cong Y^{\prime}_{k}$, the quotient
$X^{\prime}_{k}\to Y^{\prime}_{k}$ has sections. Let $\sigma:Y^{\prime}_{k}\to
X^{\prime}_{k}$ be a section. The following diagram commutes

${Y^{\prime}_{k}\times_{Y^{\prime}}Y^{\prime}_{l}}$${X^{\prime}_{k}}$${X^{\prime}}$${Y^{\prime}_{k}}$${Y^{\prime}_{l}}$${Y^{\prime}.}$$\scriptstyle{\sigma\circ\mathrm{pr}_{k}}$$\scriptstyle{\mathrm{pr}_{k}}$$\scriptstyle{\mathrm{pr}_{l}}$$\scriptstyle{\iota_{k}}$
(111)

Then there is a morphism
$\varrho:Y^{\prime}_{k}\times_{Y^{\prime}}Y^{\prime}_{k}\to
X^{\prime}_{l}=X^{\prime}\times_{Y^{\prime}}Y^{\prime}_{l}$ fitting into the
diagram

${Y^{\prime}_{k}\times_{Y^{\prime}}Y^{\prime}_{l}}$${X^{\prime}_{l}}$${X^{\prime}}$${Y^{\prime}_{l}}$${Y^{\prime}.}$$\scriptstyle{\varrho}$$\scriptstyle{\iota_{k}\circ\sigma_{k}\circ\mathrm{pr}_{k}}$$\scriptstyle{\mathrm{pr}_{l}}$${\lrcorner}$$\scriptstyle{\iota_{l}}$
(112)

Let $\sigma,\varrho$ be as above. Then the following diagram of presheaves
commutes

${Y^{\prime}_{k}\times_{Y^{\prime}}Y^{\prime}_{l}}$${Y^{\prime}_{k}}$${X^{\prime}_{k}}$${X^{\prime}_{k}/_{p}U}$${X^{\prime}_{l}}$${X^{\prime}}$${Y^{\prime}_{l}}$${X^{\prime}_{l}/_{p}U}$${X^{\prime}/_{p}U}$${(X^{\prime}/_{p}U)^{\sharp}}$$\scriptstyle{\mathrm{pr}_{k}}$$\scriptstyle{\sigma\circ\mathrm{pr}_{k}}$$\scriptstyle{\varrho}$$\scriptstyle{\mathrm{pr}_{l}}$$\scriptstyle{\cong}$$\scriptstyle{\zeta_{k}}$$\scriptstyle{\iota_{k}}$$\scriptstyle{\iota_{l}}$$\scriptstyle{\cong}$$\scriptstyle{\zeta_{l}}$
(113)

Then for $\zeta_{k}\in(X^{\prime}/_{p}U)^{\sharp}(Y^{\prime}_{k})$ and
$\zeta_{l}\in(X^{\prime}/_{p}U)^{\sharp}(Y^{\prime}_{l})$ we have

$\mathrm{pr}_{k}^{*}(\zeta_{k})=\mathrm{pr}_{l}^{*}(\zeta_{l})\in(X^{\prime}/_{p}U)^{\sharp}\big{(}Y^{\prime}_{k}\times_{Y^{\prime}}Y^{\prime}_{l}\big{)}.$
(114)

This holds for all $k,l\in J$, so $\\{\zeta_{j}\\}_{j\in J}$ is a descent
datum. Since $(X^{\prime}/_{p}U)^{\sharp}$ is a sheaf, there is a unique
$\zeta\in(X^{\prime}/_{p}U)^{\sharp}(Y^{\prime})$ such that
$\zeta|_{Y^{\prime}_{j}}=\zeta_{j}$. This element $\zeta$ does not depend on
the covering $\\{Y^{\prime}_{j}\to Y^{\prime}\\}_{j\in J}$, because $\zeta$
does not change when the covering is refined and any two coverings have a
common refinement. The element
$\zeta\in(X^{\prime}/_{p}U)^{\sharp}(Y^{\prime})$ is equivalent to a morphism
$\zeta:Y^{\prime}\to(X^{\prime}/_{p}U)^{\sharp}$ of sheaves by the Yoneda
lemma. We omit the verification that $\zeta$ is the inverse of
$(X^{\prime}/_{p}U)^{\sharp}\to Y^{\prime}$. ∎

####  3.3.3 Gluing affine quotients

We can glue the affine quotients into a quasi-projective quotient.

######  Theorem 3.24.

For the linearisation $\hat{U}\curvearrowright(X,L)$, assume Condition UU.
Then $X^{0}_{\min}$ admits a universal geometric quotient by $U$ in
$\mathrm{Sch}/\Bbbk$

$\pi:X^{0}_{\min}\to Y.$ (115)

and $\pi$ is locally isomorphic to $\mathbb{A}^{r-k}\times_{\Bbbk}Y$ over $Y$,
where $r=\dim U$ and $k\in\mathbb{N}_{+}$ is the rank of
$\mathrm{coker}(\phi)|_{X^{0}_{\min}}$.

Moreover there exists an ample line bundle $M$ on $Y$ such that $\pi^{*}M\cong
L^{m}|_{X^{0}_{\min}}$ for some $m\in\mathbb{N}_{+}$.

###### Proof.

Recall that the linearisation $\hat{U}\curvearrowright(X,L)$ satisfies
Condition UU. At the end of Section 3.3.1, we have achieved the situation
where:

  * •

There exists $m\in\mathbb{N}_{+}$ and $f_{1},\cdots,f_{t}\in
H^{0}(X,L^{m})_{\lambda=\max}$ such that $X^{0}_{\min}$ is covered by
$\\{X_{f_{i}}\\}_{i=1}^{t}$;

  * •

For each $X_{f_{i}}$, there exists a linear subspace
$\mathfrak{s}_{i}^{*}\subseteq\mathfrak{u}^{*}$ such that
$\mathfrak{s}_{i}^{*}\otimes_{\Bbbk}\mathcal{O}_{X_{f_{i}}}\cong\mathrm{coker}(\phi)|_{X_{f_{i}}}$

${\mathfrak{s}_{i}^{*}\otimes_{\Bbbk}\mathcal{O}_{X_{f_{i}}}}$${\Omega_{X_{f_{i}}/\Bbbk}}$${\mathfrak{u}^{*}\otimes_{\Bbbk}\mathcal{O}_{X_{f_{i}}}}$${\mathrm{coker}(\phi)|_{X_{f_{i}}}}$${0.}$$\scriptstyle{\subseteq}$$\scriptstyle{\cong}$$\scriptstyle{\phi|_{X_{f_{i}}}}$
(116)

We can apply Proposition 3.22 to each of $X_{f_{i}}$. For each $1\leq i\leq
i\leq t$, there exists a universal geometric quotient $\pi_{i}:X_{f_{i}}\to
Y_{i}:=\mathrm{Spec}\big{(}\mathcal{O}(X_{f_{i}})^{U}\big{)}$.

We first glue the affine schemes $Y_{i}$ for $1\leq i\leq t$. For $1\leq
i,j\leq t$, consider $f_{j}/f_{i}\in\mathcal{O}(X_{f_{i}})$. We have
$f_{j}/f_{i}\in\mathcal{O}(X_{f_{i}})^{U}$. Since $\pi_{i}:X_{f_{i}}\to Y_{i}$
is a geometric quotient by $U$, we have
$\pi_{i}^{*}:\mathcal{O}(Y_{i})\cong\mathcal{O}(X_{f_{i}})^{U}$. Then there
exists a unique $g_{ji}\in\mathcal{O}(Y_{i})$ such that
$\pi_{i}^{*}(g_{ji})=f_{j}/f_{i}$. Let
$Y_{ij}=Y_{i}\setminus\mathbb{V}(g_{ji})$ be a principal affine open
subscheme. Then $\pi_{i}^{-1}(Y_{ij})=X_{f_{i}f_{j}}=\pi_{j}^{-1}(Y_{ji})$.
Since $\pi_{i},\pi_{j}$ are universal geometric quotients, both $Y_{ij}$ and
$Y_{ji}$ are geometric quotients of $X_{f_{i}f_{j}}$. Therefore canonically
$Y_{ij}\cong Y_{ji}$ and this isomorphism is the unique morphism such that the
following diagram commutes

${X_{f_{i}f_{j}}}$${X_{f_{j}f_{i}}}$${Y_{ij}}$${Y_{ji}.}$$\scriptstyle{\pi_{i}|_{X_{f_{i}f_{j}}}}$$\scriptstyle{\pi_{j}|_{X_{f_{j}f_{i}}}}$$\scriptstyle{\cong}$
(117)

It follows easily that $\\{Y_{i}\\}_{i=1}^{t}$ glue to a scheme $Y$ containing
each $Y_{i}$ as an affine open subscheme. Moreover the morphisms
$\pi_{i}:X_{f_{i}}\to Y_{i}$ glue to a morphism

$\pi:X^{0}_{\min}\to Y$ (118)

which is an affine universal geometric quotient by $U$.

We then construct an ample line bundle $M$ on $Y$ such that $\pi^{*}M\cong
L^{m}|_{X^{0}_{\min}}$, where $m\in\mathbb{N}_{+}$ is the integer such that
$f_{i}\in H^{0}(X,L^{m})_{\lambda=\max}$. Recall functions
$g_{ji}\in\mathcal{O}(Y_{i})$ are invertible on $Y_{ij}$, i.e.
$g_{ji}|_{Y_{ij}}\in\mathcal{O}(Y_{ij})^{\times}$. They satisfy the cocycle
condition $g_{ij}g_{jk}g_{ki}=1$ in $\mathcal{O}(Y_{i}\cap Y_{j}\cap Y_{k})$
for all $1\leq i,j,k\leq t$. Then there exists a line bundle $M$ whose
transition functions are $\\{g_{ij}\\}_{i,j=1}^{t}$. It follows that
$\pi^{*}M$ and $L^{m}|_{X^{0}_{\min}}$ both have transition functions
$\\{f_{j}/f_{i}\\}_{i,j=1}^{t}$ with respect to the covering
$\\{X_{f_{i}}\\}_{i=1}^{t}$. Therefore $\pi^{*}M\cong L^{m}|_{X^{0}_{\min}}$.

It remains to show that $M$ is ample. The scheme $Y=\bigcup_{i=1}^{t}Y_{i}$ is
quasi-compact since it is a finite union of affine schemes. We will construct
sections $g^{(i)}\in\Gamma(Y,M)$ for $1\leq i\leq t$ such that each
$Y_{g^{(i)}}=Y_{i}$ is affine. Fix $i$. Consider
$g^{(i)}_{j}:=g_{ji}\in\mathcal{O}(Y_{j})$ for $1\leq j\leq t$. Then
$\\{g^{(i)}_{j}\\}_{j=1}^{t}$ define a global section $g^{(i)}\in\Gamma(Y,M)$
since

$g^{(i)}_{j_{1}}=g_{j_{1}j_{2}}g^{(i)}_{j_{2}},\quad\textrm{ for }1\leq
j_{1},j_{2}\leq t.$ (119)

For each $j$, we have

$Y_{j}\cap Y_{g^{(i)}}=(Y_{j})_{g^{(i)}_{j}}=(Y_{j})_{g_{ji}}=Y_{ji}=Y_{j}\cap
Y_{i}$ (120)

which proves that $Y_{g^{(i)}}=Y_{i}$. This proves that $M$ is ample
(Definition 01PS). ∎

There is a more concrete description of an immersion of the quotient
$Y=X^{0}_{\min}/U$ into a projective space.

######  Lemma 3.25.

Assume Condition UU. Let $\pi:X^{0}_{\min}\to Y$ be the quotient in Theorem
3.24. Then for sufficiently divisible $m\in\mathbb{N}_{+}$, there exists a
commutative diagram

${X}$${\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}}$${X^{0}_{\min}}$${\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}^{0}_{\min}}$${Y}$${\mathbb{P}\Big{(}\big{(}H^{0}(X,L^{m})^{U}\big{)}^{*}\Big{)}^{0}_{\min}}$$\scriptstyle{\subseteq}$$\scriptstyle{\pi}$$\scriptstyle{\subseteq}$
(121)

###### Proof.

According to the proof of Theorem 3.24, there exists $m_{1}\in\mathbb{N}_{+}$
such that:

  * •

$H^{0}(X,L^{m_{1}})$ generates $\bigoplus_{d\geq 0}H^{0}(X,L^{dm_{1}})$;

  * •

there exist $t\in\mathbb{N}_{+}$ and $f_{1},\cdots,f_{t}\in
H^{0}(X,L^{m_{1}})_{\lambda=\max}$;

  * •

$X^{0}_{\min}=\bigcup_{i=1}^{t}X_{f_{i}}$;

  * •

for each $1\leq i\leq t$, the invariant subring $\mathcal{O}(X_{f_{i}})^{U}$
is finitely generated.

Then there exists $m_{2}\in\mathbb{N}_{+}$ such that for each $1\leq i\leq t$,
we can choose $n_{i}\in\mathbb{N}_{+}$ generators of
$\mathcal{O}(X_{f_{i}})^{U}$ of the form

$\frac{g^{(i)}_{j}}{f_{i}^{m_{2}}}\in\mathcal{O}(X_{f_{i}})^{U}\quad
j=1,\cdots,n_{i}$ (122)

for $g^{(i)}_{j}\in H^{0}(X,L^{m_{1}m_{2}})$. Since these finitely many
generators $\frac{g^{(i)}_{j}}{f_{i}^{m_{2}}}$ are invariant under
$\mathfrak{u}$ and $\dim\mathfrak{u}<\infty$, there exists
$m_{3}\in\mathbb{N}_{+}$ such that for all $i,j$ we have
$f_{i}^{m_{3}}g^{(i)}_{j}\in H^{0}(X,L^{m_{1}(m_{2}+m_{3})})^{U}$.

Let $m=m_{1}(m_{2}+m_{3})\in\mathbb{N}_{+}$. The above shows that for each
$1\leq i\leq t$, there is a surjective map

$\frac{H^{0}(X,L^{m})^{U}}{f_{i}^{m_{2}+m_{3}}}\to\mathcal{O}(X_{f_{i}})^{U}$
(123)

and then the following ring map is surjective for $1\leq i\leq t$

$\Big{(}\mathrm{Sym}^{\bullet}H^{0}(X,L^{m})^{U}\Big{)}_{(f_{i}^{m_{2}+m_{3}})}\to\mathcal{O}(X_{f_{i}})^{U}$
(124)

which corresponds to a closed immersion

$X_{f_{i}}/U\hookrightarrow\mathbb{P}\Big{(}\big{(}H^{0}(X,L^{m})^{U}\big{)}^{*}\Big{)}_{f_{i}^{m_{2}+m_{3}}}.$
(125)

By the construction of $\pi:X^{0}_{\min}\to Y$, we have that $Y$ is glued from
$X_{f_{i}}/U$. The above closed immersions for $1\leq i\leq t$ glue to a
closed immersion

$Y\hookrightarrow\mathbb{P}\Big{(}\big{(}H^{0}(X,L^{m})^{U}\big{)}^{*}\Big{)}^{0}_{\min}.$
(126)

The closed immersions
$X\hookrightarrow\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}$ and
$X^{0}_{\min}\hookrightarrow\mathbb{P}\big{(}H^{0}(X,L^{m})^{*}\big{)}^{0}_{\min}$
are naturally defined. It is easy to see these three closed immersions fit
into the commutative diagram as stated. ∎

######  Corollary 3.26.

For the linearisation $\hat{U}\curvearrowright(X,L)$, assume Condition UU. Let
$\pi:X^{0}_{\min}\to Y$ be the quotient in Theorem 3.24. Then $Y$ represents
the quotient sheaf $(X^{0}_{\min}/_{p}U)^{\sharp}$ on
$(\mathrm{Sch}/\Bbbk)_{\acute{e}tale}$ (Definition 03BD).

Moreover, for any morphism $Y^{\prime}\to Y$ in $\mathrm{Sch}/\Bbbk$, the
quotient sheaf $\big{(}X^{\prime}/_{p}U\big{)}^{\sharp}$ is represented by
$Y^{\prime}$, where $X^{\prime}:=X^{0}_{\min}\times_{Y}Y^{\prime}$.

###### Proof.

According to the proof of Theorem 3.24, the quotient $\pi:X^{0}_{\min}\to Y$
is glued from $\pi|_{X_{f_{i}}}:X_{f_{i}}\to Y_{i}:=X_{f_{i}}/U$ for some
$m\in\mathbb{N}_{+}$ and $f_{1},\cdots,f_{t}\in
H^{0}(X,L^{m})_{\lambda=\max}$. Let $\\{Y_{j}\to Y\\}_{j\in J}$ denote this
Zariski covering. which is also a covering in
$(\mathrm{Sch}/\Bbbk)_{\acute{e}tale}$. Let $\\{X_{j}\to X^{0}_{\min}\\}_{j\in
J}$ be the base change covering of $\\{Y_{j}\to Y\\}$. By Corollary 3.23, we
have $(X_{j}/_{p}U)^{\sharp}\cong Y_{j}$ for all $j\in J$, and the
representability is stable under base change, in the sense that
$\big{(}X_{j}\times_{Y_{j}}W\big{/}_{p}U\big{)}^{\sharp}\cong W$ for any
morphism $W\to Y_{j}$ in $\mathrm{Sch}/\Bbbk$.

Let $Y^{\prime}\to Y$ be a morphism in $\mathrm{Sch}/\Bbbk$. Denote

$X^{\prime}:=X\times_{Y}Y^{\prime},\quad
Y^{\prime}_{j}:=Y_{j}\times_{Y}Y^{\prime},\quad
X^{\prime}_{j}:=X^{\prime}\times_{Y^{\prime}}Y^{\prime}_{j}.$ (127)

Then $X^{\prime}\to Y^{\prime}$ is a geometric quotient as a base change of a
universal geometric quotient. The quotient induces a morphism
$(X^{\prime}/_{p}U)^{\sharp}\to Y^{\prime}$ of sheaves. We will construct an
element $\zeta\in(X^{\prime}/_{p}U)^{\sharp}(Y^{\prime})$ such that the
corresponding morphism $Y^{\prime}\to(X^{\prime}/_{p}U)^{\sharp}$ is inverse
to $(X^{\prime}/_{p}U)^{\sharp}\to Y^{\prime}$.

By Corollary 3.23, we have $(X^{\prime}_{j}/_{p}U)^{\sharp}\cong
Y^{\prime}_{j}$, which is the base change of $(X_{j}/_{p}U)^{\sharp}\cong
Y_{j}$ along $Y^{\prime}_{j}\to Y_{j}$. The morphism
$X^{\prime}_{j}\to(X^{\prime}_{j}/_{p}U)^{\sharp}$ is surjective as a morphism
of sheaves, so there exists a covering
$\\{\varphi_{j}:\widetilde{Y^{\prime}_{j}}\to Y^{\prime}_{j}\\}$ in
$(\mathrm{Sch}/\Bbbk)_{\acute{e}tale}$ and a morphism
$\sigma_{j}:\widetilde{Y^{\prime}_{j}}\to X^{\prime}_{j}$ fitting into the
diagram

${\widetilde{Y^{\prime}_{j}}}$${X^{\prime}_{j}}$${Y^{\prime}_{j}}$${(X^{\prime}_{j}/_{p}U)^{\sharp}.}$$\scriptstyle{\sigma_{j}}$$\scriptstyle{\varphi_{j}}$$\scriptstyle{\cong}$
(128)

The composition of $\\{\varphi_{j}:\widetilde{Y^{\prime}_{j}}\to
Y^{\prime}_{j}\\}$ and $\\{Y^{\prime}_{j}\to Y^{\prime}\\}$ is also a covering
$\\{\widetilde{Y^{\prime}_{j}}\to Y^{\prime}\\}_{j\in J}$ in
$(\mathrm{Sch}/\Bbbk)_{\acute{e}tale}$. Since $(X^{\prime}/_{p}U)^{\sharp}$ is
a sheaf, the following is an equaliser diagram