# GOOD MODELS OF HILBERT SCHEMES OF POINTS OVER SEMISTABLE DEGENERATIONS

CALLA TSCHANZ

Abstract. In this paper, we explore different possible choices of expanded
degenerations and define appropriate stability conditions in order to
construct good degenerations of Hilbert schemes of points over semistable
degenerations of surfaces, given as proper Deligne-Mumford stacks. These
stacks provide explicit examples of constructions arising from the work of
Maulik and Ranganathan. This paper builds upon and generalises previous work
in which we constructed a special example of such a stack. We also explain how
these methods apply to constructing minimal models of type III degenerations
of hyperkähler varieties, namely Hilbert schemes of points on K3 surfaces.

###### Contents

  1. 1 Introduction
  2. 2 Background
  3. 3 The expanded constructions
  4. 4 The stack construction
  5. 5 Constructing separated stacks.
  6. 6 Hilbert schemes of points on K3 surfaces and MMP

##  1 Introduction

An important method in the study of smooth varieties is to allow these
varieties to degenerate to a sum of simpler irreducible components and use the
degeneration to recover information about the original variety. Our aim is to
explore various constructions of good degenerations of Hilbert schemes of
points over a semistable family of surfaces $X\to C$. We combine techniques of
expanded degenerations (or expansions), geometric invariant theory, stacks and
tropical geometry, as well as observe a symmetry breaking phenomenon in the
constructions which allows us to define explicit examples. This builds upon
previous work in which we constructed such a degeneration of Hilbert schemes
of points as a proper Deligne-Mumford stack ([Tsc23]). The stability
conditions required to make this stack proper are much simpler than would be
expected in general for this type of problem. This is an interesting
consequence of the specific choices made in the expanded degenerations we
considered. In this paper, we generalise this construction, investigate
alternative choices of expanded degenerations and describe the appropriate
refinements of the stability condition needed to obtain proper moduli stacks.
We give a detailed analysis of the stability conditions needed to ensure
universal closure and separatedness, both from a geometric and tropical
perspective.

#### Setup and motivation.

We recall the basic setup of this problem from [Tsc23]. Let $X\to C$ be a
semistable projective family of surfaces over a smooth curve such that the
singularities of the special fibre $X_{0}$ over $0\in C$ are given étale
locally by three planes intersecting transversely in $\mathbb{A}^{3}$. In
other words, we may consider the étale local model for this family given by
$\operatorname{Spec}k[x,y,z,t]/(xyz-t)$, where $k$ is an algebraically closed
field of characteristic zero.

Now, let $X^{\circ}\coloneqq X\setminus X_{0}$ and $C^{\circ}\coloneqq
C\setminus\\{0\\}$ and consider the relative Hilbert scheme of $m$ points
$\operatorname{Hilb}^{m}(X^{\circ}/C^{\circ})$. Our goal is to construct
compactifications of this object which satisfy some desired properties, which
we discuss in the two bullet points below. The obvious compactification
$\operatorname{Hilb}^{m}(X/C)$ does not satisfy these properties. In the case
of the first bullet point, this comes from the fact that
$\operatorname{Hilb}^{m}(X/C)$ is not logarithmically flat with respect to the
pullback divisorial logarithmic structure given by the divisor $X_{0}$ in $X$.
For the second point, it follows from the fact that the special fibre of
$\operatorname{Hilb}^{m}(X/C)$ is very singular (which makes the total space
very singular).

In $\operatorname{Hilb}^{m}(X^{\circ}/C^{\circ})$, we have removed the limits
of the families of length $m$ zero-dimensional subschemes and we wish to
compactify, i.e. add limits back in in such a way that the resulting object
satisfies “good” properties. These limits will again be length $m$ zero-
dimensional subschemes, some on $X_{0}$ and some on different varieties. This
will be achieved by constructing _expansions_ (birational modifications of
$X_{0}$ in a 1-parameter family) and selecting the appropriate limit for each
family of length $m$ zero-dimensional subschemes in these modifications with
the help of a stability condition. The two main meanings of “good” we may
consider are the following.

  * •

The resulting degeneration of Hilbert schemes of points should be flat and
proper over $C$, and the limit over $0\in C$ of each family of length $m$
zero-dimensional subschemes should be supported in the smooth locus of an
underlying surface. Clearly, this is not the case for the obvious
compactification $\operatorname{Hilb}^{m}(X/C)$, as certain length $m$ zero-
dimensional subschemes in the fibre over $0\in C$ will have some of their
support lying in $\operatorname{Sing}(X_{0})$. We wish to remove these
subschemes and replace them by subschemes having support in the smooth locus
of some modifications of $X_{0}$. As in [Tsc23], we will refer to the support
of a subscheme lying in the smooth locus of a surface as the condition that
the subscheme is _smoothly supported_. Imposing these properties has the
following useful consequences. The question of studying Hilbert schemes of $m$
points on $X_{0}$ is replaced by an easier problem. We study the Hilbert
schemes of $m^{\prime}$ points on the interior of each irreducible component
of the modifications of $X_{0}$, where $m^{\prime}\leq m$, and consider
products of these simpler Hilbert schemes of points. The smooth fibres of the
family of Hilbert schemes of $m$ points degenerate to a union of these
products, which is easier to describe. Moreover, in such a compactification,
the special fibre is stratified in a way which encodes the data of the
families of length $m$ zero-dimensional subschemes. By observing where the
limit of a family lands within this stratification, we may recover the
defining equations of this family. These properties have applications to
cohomological computations and enumerative geometry.

  * •

Secondly, one can consider the more specific case where $X\to C$ is a type III
semistable degeneration of K3 surfaces, with a relative holomorphic symplectic
logarithmic 2-form. We can then try to construct a family of Hilbert schemes
of points on $X$ which will be minimal in the sense of the minimal model
program, and by this we mean a semistable or divisorial log terminal (dlt)
degeneration with trivial canonical bundle (see Section 6 for details on
minimality). The singularities arising in such a degeneration $X\to C$ are of
the type described above, i.e. we can restrict ourselves to the local problem
where $X_{0}$ is thought of as given by $xyz=0$ in $\mathbb{A}^{3}$. Among
other reasons, Hilbert schemes of points on K3 surfaces are interesting to
study because they form a class of examples of hyperkähler varieties.

In this paper, we will construct degenerations of Hilbert schemes of points
satisfying the properties of the first point above. We then show that the
solutions considered here also give us constructions of semistable minimal
hyperkähler degenerations as stacks. It is worth noting, however, that it is
possible to construct dlt minimal degenerations as schemes which do not
fulfill the requirement of the first point.

#### Main results.

Here, we briefly describe the main results of this paper.

We start by making a sequence of blow-ups on a fibre product
$X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ over a base
$C\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$. This process is similar to the
construction in [Tsc23] (see Section 2.1 for an overview of this previous
construction), however, here, we allow for more types of blow-ups to be made.
The sequence of blow-ups on $X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ is
encoded by a choice of sets $A,B\subset\\{1,\dots,n+1\\}$ which satisfy a
specific property called the _unbroken_ property (see Definition 3.1). We name
the resulting expanded degenerations $X[A,B]$. For a specific choice of $A$
and $B$, we recover the expanded degeneration $X[n]$ of [Tsc23]. We describe
an action of $\mathbb{G}_{m}^{n}$ on $X[A,B]$, an embedding of $X[A,B]$ into a
product of projective bundles and discuss various GIT stability conditions on
the relative Hilbert scheme of $m$ points of this expanded degeneration
$\operatorname{Hilb}^{m}(X[A,B]/C\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1})$. We
then present a stack of expansions $\mathfrak{C}^{\prime}$ with a family
$\mathfrak{X}^{\prime}$ over it and explore the different choices of stability
condition which will allow us to construct a stack of stable length $m$ zero-
dimensional subschemes in $\mathfrak{X}^{\prime}$ which is proper over $C$. We
start by extending the LW and SWS stability conditions defined in [Tsc23] to
this setting and denote by $\mathfrak{N}^{m}_{\operatorname{LW}}$ and
$\mathfrak{N}^{m}_{\operatorname{SWS}}$ the corresponding stacks of LW and SWS
stable length $m$ zero-dimensional subschemes on $\mathfrak{X}^{\prime}$. We
show that, though both of these stacks are universally closed, they are not
separated, as they contain several non-equivalent representatives for the same
limit. We consider two main approaches to solve this problem.

The first method to construct separated stacks is to define equivalence
classes identifying all limits of a given object in
$\mathfrak{N}^{m}_{\operatorname{LW}}$ and
$\mathfrak{N}^{m}_{\operatorname{SWS}}$. The stacks whose objects are these
equivalence classes are denoted
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}$ and
$\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}$. These are no longer
algebraic stacks but we may now prove the following theorem.

######  Theorem 1.1.

The stacks $\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}$ and
$\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}$ are proper over $C$ and
have finite automorphisms.

This approach parallels work of Kennedy-Hunt on logarithmic Quot schemes
[Ken23].

The second method we use is to define an additional stability condition,
called $(\alpha,\beta)$-stability, which we use to cut out proper substacks of
$\mathfrak{N}^{m}_{\operatorname{LW}}$ and
$\mathfrak{N}^{m}_{\operatorname{SWS}}$. The $(\alpha,\beta)$-stability
condition must satisfy some subtle properties in order for the substack it
cuts out to be proper over $C$: it must be what we call a proper LW or SWS
stability condition (see Definition 5.10). A proper LW stability condition
$(\alpha,\beta)$ as defined here recovers one of the choices of stability
condition arising from the methods of Maulik and Ranganathan [MR24] in the
case of Hilbert schemes of points. We denote by
$\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}$ and
$\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}$ the stacks of stable
length $m$ zero-dimensional subschemes in $\mathfrak{X}^{\prime}$ for proper
LW and SWS stability conditions $(\alpha,\beta)$. We are then able to prove
the following results.

######  Theorem 1.2.

The stacks $\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}$ and
$\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}$ are Deligne-Mumford
and proper over $C$.

######  Theorem 1.3.

For any $(\alpha,\beta)$ which defines a proper SWS stability condition, there
is an isomorphism of stacks

$\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}\cong\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}.$

We would like to emphasize here that it is not true that for every stack
$\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}$ there exists an
equivalent stack $\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}$.

These choices of proper substack distinguish some geometrically meaningful
examples among all possible choices arising from the methods of Maulik and
Ranganathan. Our examples do not recover every possible choice, because the
geometric assumptions made in the construction of $\mathfrak{X^{\prime}}$ mean
that it does not contain all possible expansions of $X$. The reason for these
assumptions is discussed in Section 3.1. The stacks
$\mathfrak{M}^{m}_{\operatorname{LW}}$ and
$\mathfrak{M}^{m}_{\operatorname{SWS}}$ of [Tsc23] are special cases of
$\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}$ and
$\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}$ stacks.

An important concern in the construction of these proper stacks is to
understand what smoothings exist from one special element of the degeneration
to another. Indeed, we must ensure that not only the generic elements of a
family have unique limits, but also that special elements of these families
with support in modifications of $X_{0}$ have unique limits as well. In order
to show that this is the case, it is necessary to study what smoothings exist
from one modification of $X_{0}$ to another. We give a tropical criterion to
explain how to understand these smoothings in Section 5.4.

Finally, we are able to show that all proper algebraic stacks constructed here
provide good minimal models (see Section 6 for a precise definition of the
term good minimal model). In particular, we prove the following result.

######  Theorem 1.4.

If $X\to C$ is a semistable type III degeneration of K3 surfaces equipped with
a relative holomorphic symplectic logarithmic 2-form, then this 2-form induces
a nowhere degenerate relative logarithmic 2-form on each of the constructions
presented above.

#### Background.

Previous work in this area includes that of Li [Li13], in which the method of
expanded degenerations is introduced, and Li and Wu [LW15], where it is used
to construct modular compactifications for degenerations of Quot schemes
relative to simple normal crossing boundaries with smooth singular locus.
Following on from this, Gulbrandsen, Halle and Hulek [GHH19] present a GIT
version in the case of Hilbert schemes of points, and show the stack quotient
of their GIT quotient to be equivalent to the proper Deligne-Mumford stack of
Li and Wu. These solutions provide a foundation for our work, but are limited
by the requirement that the special fibre of $X\to C$ should have smooth
singular locus. This is quite a serious restriction, as not many families will
satisfy this condition. Note that the type of singularity we consider in this
work is specifically the type that [LW15] and [GHH19] do not address.

In more recent work, Maulik and Ranganathan [MR24] use these techniques of
expansions in the world of logarithmic and tropical geometry to solve this
problem for Hilbert schemes over $X$ where $X_{0}$ may have any type of simple
normal crossing singularity. This is a significant improvement, as their
solution can be applied to any semistable family of surfaces $X$ and these can
always be obtained by semistable reduction. Their construction, however,
yields an infinite family of birational solutions, but no explicit model.
Moreover, it is far from obvious how such an explicit model should be
constructed. This is the problem we solve in [Tsc23] and in this paper.

#### Organisation.

We will start by giving a brief overview of the results of [Tsc23] and by
setting up the tropical point if view in Section 2. In Section 3, we construct
the expanded degenerations $X[A,B]$ and extend the group action, line bundle
and GIT discussions from [Tsc23] to this context. Then, in Section 4 we define
the stack of expansions $\mathfrak{C}^{\prime}$, the family
$\mathfrak{X}^{\prime}$ over $\mathfrak{C}^{\prime}$. We describe how LW and
SWS stability extend to this context, define the corresponding stacks
$\mathfrak{N}^{m}_{\operatorname{LW}}$ and
$\mathfrak{N}^{m}_{\operatorname{SWS}}$ of stable subschemes on
$\mathfrak{X}^{\prime}$ and show that these are universally closed but not
separated. In Section 5, we discuss different methods for constructing
separated stacks, as well as a tropical criterion for understanding smoothings
between modifications of $X_{0}$. Finally, we discuss how all these
constructions relate to the minimal model program in Section 6.

#### Acknowledgements.

I am grateful to Gregory Sankaran for his support and encouragement during
this project. I would also like to thank Dhruv Ranganathan, Alastair Craw,
Patrick Kennedy-Hunt and Thibault Poiret for all their help, as well as Siao
Chi Mok and Terry Song for many interesting conversations. I am also grateful
to Christian Lehn, Martin Ulirsch, Grégoire Menet and Grzegorz Kapustka for
their useful suggestions. This work was undertaken while funded by the
University of Bath Research Studentship Award and completed while funded under
the grant Narodowe Centrum Nauki 2018/30/E/ST1/00530 at the Jagiellonian
University in Krakow.

##  2 Background

In this section we recall the main results of [Tsc23] and the setup of the
tropical perspective which we will refer to throughout this paper.

#### The semistable family of surfaces.

Let $X\to C$ be a flat projective semistable family of surfaces over a curve
isomorphic to $\mathbb{A}^{1}$, and we consider an étale local model
$\operatorname{Spec}k[x,y,z,t]/(xyz-t)\to\operatorname{Spec}k[t]$ for this
family, where $k$ is an algebraically closed field of characteristic zero. We
denote by $X_{0}$ the special fibre over $0\in C$ and by $Y_{1}$, $Y_{2}$ and
$Y_{3}$ the irreducible components of this special fibre given étale locally
by $x=0,y=0$ and $z=0$ respectively.

###  2.1 The previous construction

In [Tsc23], we start by constructing expanded degenerations $X[n]$ given by a
sequence of blow-ups on a fibre product
$X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ over an expanded base
$C[n]\coloneqq C\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$. This construction is
similar to those presented in [LW15] and [GHH19], but additional blow-ups are
added, breaking the symmetry of the construction, to allow for all necessary
modifications for our problem to be included, while preserving commutativity
of the blow-up morphisms. This family $X[n]$ comes with a natural torus action
by the group $\mathbb{G}_{m}^{n}$ and is shown to embed into a product of
projective bundles. This allows us to define various
$\mathbb{G}_{m}^{n}$-linearised ample line bundles on the relative Hilbert
scheme of $m$ points of this expanded degeneration
$\operatorname{Hilb}^{m}(X[n]/C[n])$ and describe the corresponding GIT stable
loci.

We then construct a stack of expansions $\mathfrak{C}$ and family
$\mathfrak{X}$ over it. This is a substack of the stack
$\mathfrak{X}^{\prime}$ defined in Section 4.2. We describe how Li-Wu
stability (abbreviated LW stability) can be extended to this setting and
define an alternative notion of stability, called smoothly supported weak
strict stability (abbreviated SWS stability), derived from GIT stability
conditions. We then construct the stacks
$\mathfrak{M}^{m}_{\operatorname{LW}}$ and
$\mathfrak{M}^{m}_{\operatorname{SWS}}$ of LW and SWS stable length $m$ zero-
dimensional subschemes on $\mathfrak{X}$. The main results of [Tsc23] are the
following.

######  Theorem 2.1.

The stacks $\mathfrak{M}^{m}_{\operatorname{LW}}$ and
$\mathfrak{M}^{m}_{\operatorname{SWS}}$ are Deligne-Mumford and proper.

######  Theorem 2.2.

There is an isomorphism of stacks

$\mathfrak{M}^{m}_{\operatorname{LW}}\cong\mathfrak{M}^{m}_{\operatorname{SWS}}.$

Crucially, the choices of blow-ups allowed in the construction of $X[n]$ are
very restricted. This has the surprising consequence that the stacks of LW and
SWS stable subschemes $\mathfrak{M}^{m}_{\operatorname{LW}}$ and
$\mathfrak{M}^{m}_{\operatorname{SWS}}$ are already proper without the need to
add any additional stability condition. The work of Maulik and Ranganathan
[MR24] predicts that in general this will not be the case. Indeed, we will see
in Section 5 that when allowing for different choices of expanded
degenerations we will need to introduce an additional stability condition in
order to obtain separated stacks over $C$. This stability condition is
analogous to the Donaldson-Thomas stability of [MR24].

###  2.2 Tropical perspective

Throughout this work, the results will be presented from a geometric and
tropical point of view. Indeed, the key property that we would like our
compactifications of $\operatorname{Hilb}^{m}(X^{\circ}/C^{\circ})$ to possess
can be thought of as the property that the limits of the families of length
$m$ zero-dimensional subschemes in the compactification should be stratified
in a way which records the data of these families. In étale local coordinates,
this is essentially asking for the compactifications to encode the degrees of
vanishing of these families in the variables $x,y$ and $z$. Tropical geometry
is a natural tool to use to study such questions, as it is precisely the
object parameterising such data.

We recall here the essential details of the tropical perspective (see [Tsc23]
and [MR24] for more details). We construct the tropicalisation of $X$ with
respect to the divisorial logarithmic structure given by the divisor $X_{0}$.
Formally, this means that we define a sheaf of monoids

$\mathcal{M}_{X}(U)\coloneqq\\{f\in\mathcal{O}_{X}(U)\ |\ f|_{U\setminus
X_{0}}\in\mathcal{O}^{*}_{X}(U\setminus X_{0})\\},$

for any open set $U\subseteq X$, and the corresponding _characteristic sheaf_
$\overline{\mathcal{M}}_{X}\coloneqq\mathcal{M}_{X}/\mathcal{O}^{*}_{X}.$ The
_tropicalisation_ of $X$ is then given by
$\operatorname{trop}(X)\coloneqq\operatorname{colim}_{x\in
X}(\overline{\mathcal{M}}_{X,x})^{\vee}$. We recall also the following
definition.

######  Definition 2.3.

Let $\Upsilon$ be a fan, let $|\Upsilon|$ be its support and $\upsilon$ be a
continuous map

$\upsilon\colon|\Upsilon|\longrightarrow\Sigma_{X}$

such that the image of every cone in $\Upsilon$ is contained in a cone of
$\Sigma_{X}$ and that is given by an integral linear map when restricted to
each cone in $\Upsilon$. We say that $\upsilon$ is a subdivision if it is
injective on the support of $\Upsilon$ and the integral points of the image of
each cone $\tau\in\Upsilon$ are exactly the intersection of the integral
points of $\Sigma_{X}$ with $\tau$.

######  Definition 2.4.

The subdivision
$\Upsilon\longhookrightarrow\Sigma_{X}\longhookrightarrow\mathbb{R}^{r}_{\geq
0}$ has an associated toric variety $\mathbb{A}_{\Upsilon}$, which comes with
a $\mathbb{G}^{r}_{m}$-equivariant birational map
$\mathbb{A}_{\Upsilon}\to\mathbb{A}^{r}$. There is then an induced morphism of
quotient stacks
$[\mathbb{A}_{\Upsilon}/\mathbb{G}_{m}^{r}]\longrightarrow[\mathbb{A}^{r}/\mathbb{G}_{m}^{r}].$
We define the expansion of $X$ associated to such a subdivision $\Upsilon$ to
be the modification

$X_{\Upsilon}\coloneqq
X\times_{[\mathbb{A}^{r}/\mathbb{G}_{m}^{r}]}[\mathbb{A}_{\Upsilon}/\mathbb{G}_{m}^{r}].$

In the étale local model which we consider, the functions vanishing at $X_{0}$
are $x,y$ and $z$, therefore we may represent $\operatorname{trop}(X)$ as a
fan in $\mathbb{R}_{\geq 0}^{3}$ given by the positive orthant and its faces.
For convenience, we will denote by $\operatorname{trop}(X_{0})$ the hyperplane
slice through $\operatorname{trop}(X)$ resulting from fixing a height in this
three-dimensional cone. Recall from [Tsc23] that this choice of height is
equivalent to a choice of point in the half-line corresponding to the
tropicalisation of $\mathbb{A}^{1}$ (where this tropicalisation is taken with
respect to the divisor $0\in\mathbb{A}^{1}$). Changing the height corresponds
to making some base change on the family $X$. Figure 1 shows a copy of the
special fibre $X_{0}$ both from the geometric point of view, on the left, and
tropical point of view, on the right. We will abuse notation slightly and
label the vertices, edges and interior of the triangle
$\operatorname{trop}(X_{0})$ by $Y_{i}$, $Y_{i}\cap Y_{j}$ and $Y_{1}\cap
Y_{2}\cap Y_{3}$ again for convenience, as shown in Figure 1. Subdivisions of
the tropicalisation will be represented by adding edges and vertices to this
triangle, as we will see in the following section.

$Y_{1}$$Y_{2}$$Y_{3}$$Y_{1}\cap Y_{2}$$Y_{1}\cap Y_{3}$$Y_{2}\cap
Y_{3}$$Y_{1}\cap Y_{2}\cap Y_{3}$

$Y_{1}\cap Y_{2}$$Y_{2}\cap Y_{3}$$Y_{1}\cap
Y_{3}$$Y_{1}$$Y_{3}$$Y_{2}$$Y_{1}\cap Y_{2}\cap Y_{3}$

Figure 1: Geometric and tropical pictures of the special fibre $X_{0}$.

Finally, we consider the _tropicalisation map_ , as defined in [MR24]. This is
a morphism

$\operatorname{trop}\colon X^{\circ}\longrightarrow\Sigma_{X}.$

For an open subscheme $Z^{\circ}\subset X^{\circ}$, we denote by
$\operatorname{trop}(Z^{\circ})$ the image of the map $\operatorname{trop}$
restricted to $Z^{\circ}(\mathcal{K})$, where $\mathcal{K}$ is a valued field
extending $k$. This image $\operatorname{trop}(Z^{\circ})$ can be seen as a
collection of rays in $\operatorname{trop}(X)$ or vertices in
$\operatorname{trop}(X_{0})$. This is the basic data around which the
expansions are built.

##  3 The expanded constructions

###  3.1 Scheme construction

We construct expanded degenerations as schemes $X[A,B]$ depending on a choice
of sets $A$ and $B$. This forms a generalisation of the schemes $X[n]$
presented in [Tsc23]. We recall some of the notation and terminology from
[Tsc23] here for convenience.

#### Enlarging the base.

Take a copy of $\mathbb{A}^{n+1}$, with elements labelled
$(t_{1},\ldots,t_{n+1})\in\mathbb{A}^{n+1}.$ To match our previous
terminology, we refer to the entries $t_{i}$ as basis directions. We then take
a fibre product $X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ given by the map
$X\to C\cong\mathbb{A}^{1}$ and the product

$(t_{1},\ldots,t_{n+1})\longmapsto t_{1}\cdots t_{n+1}.$

This fibre product contains several copies of the special fibre $X_{0}$. We
will now be able to make different modifications of this fibre by taking a
sequence of blow-ups on $X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ along
$Y_{1}$ and $Y_{2}$ and the vanishing of some basis directions. Unlike the
construction of [Tsc23], we do not require here that both $Y_{1}$ and $Y_{2}$
should be blown up along all basis directions. Instead, the choice of blow-ups
will be dictated by a choice of sets $A$ and $B$.

#### The sets $A$ and $B$.

For geometric reasons explained in Remark 3.7, we will consider blow-ups
arising only from sets $A$ and $B$ satisfying the following condition.

######  Definition 3.1.

Let $A$ and $B$ be subsets of $[n+1]\coloneqq\\{1,\dots,n+1\\}$. We say that
the pair $(A,B)$ is _unbroken_ if there are real intervals $[1,a)$ and
$(b,n+1]$ with $1<b<a<n+1$ such that $A=\mathbb{N}\cap[1,a)$ and
$B=\mathbb{N}\cap(b,n+1]$. We will often write $(A,B,n)$ for convenience when
discussing these unbroken pairs; this is done in order to keep track of the
value of $n$, although this value is determined by the sets $A$ and $B$.

In other words, if $(A,B)$ is an unbroken pair, then elements of $[n+1]$ in
order from 1 to $n+1$ are contained first in $A\setminus B$, then in $A\cap B$
and then in $B\setminus A$, where $A\cap B$ may be empty but the other two are
not. In particular, $A$ is forced to contain 1 but not $n+1$ and, similarly,
$B$ must contain $n+1$ but not 1. See Remark 3.8 for a tropical description of
the unbroken condition.

#### The blow-ups.

Let $(A,B,n)$ be an unbroken pair. In the étale local model, the blow-ups on
$X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ are expressed as blow-ups in the
ideals

$\displaystyle\langle x,t_{1}\rangle,\langle x,t_{1}t_{2}\rangle,\dots,\langle
x,t_{1}t_{2}\cdots t_{\lfloor a\rfloor}\rangle$

and the ideals

$\displaystyle\langle y,t_{n+1}\rangle,\langle
y,t_{n+1}t_{n}\rangle,\dots,\langle y,t_{n+1}t_{n}\cdots t_{\lceil
b\rceil}\rangle.$

Globally on $X$, these are blow-ups along the vanishing equations of the
components $Y_{1}$ and $Y_{2}$. The equations of the blow-ups in étale local
coordinates are given by

$\displaystyle x_{0}^{(1)}t_{1}=xx_{1}^{(1)},$ $\displaystyle
x_{1}^{(k-1)}x_{0}^{(k)}t_{k}=x_{0}^{(k-1)}x_{1}^{(k)},\qquad\textrm{ for }\
2\leq k\leq n,$ $\displaystyle x_{0}^{(\lfloor a\rfloor)}yz=x_{1}^{(\lfloor
a\rfloor)}t_{\lfloor a\rfloor+1}\cdots t_{n+1}.$ (3.1.1) $\displaystyle
y_{0}^{(1)}t_{n+1}=yy_{1}^{(1)},$ $\displaystyle
y_{1}^{(k-1)}y_{0}^{(k)}t_{n+2-k}=y_{0}^{(k-1)}y_{1}^{(k)}\ \textrm{ for }\
2\leq k\leq n,$ $\displaystyle y_{0}^{(\lceil b\rceil)}xz=y_{1}^{(\lceil
b\rceil)}t_{1}\cdots t_{\lceil b\rceil-1}$

and we get

$x_{0}^{(k)}y_{0}^{(n+1-k)}z=x_{1}^{(k)}y_{1}^{(n+1-k)}$

when $k\in A$ and $k+1\in B$.

_Expanded degeneration notation._ We denote the resulting space after making
these blow-ups by $X[A,B]$ and write $C[A,B]$ to refer to the base
$C\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ together with the data of the sets
$A$ and $B$. This is the same scheme as $C[n]$ from [Tsc23], but here, for
each basis direction $t_{i}$, we retain the additional information of whether
$i$ belongs to the set $A$ or $B$. Let

$b\colon X[A,B]\longrightarrow X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$

denote the sequence of blow-ups and

$\pi\colon X[A,B]\longrightarrow X$

denote the natural projection. If $A=\\{1,\dots,n\\}$ and
$B=\\{2,\dots,n+1\\}$, then we get back exactly the space $X[n]$ of [Tsc23].

We fix the notation $[k]\coloneqq\\{1,\dots,k\\}$. Let $|A|=l_{A}$,
$|B|=l_{B}$ and let

$\iota_{A}\colon[l_{A}]\longrightarrow
A\quad\textrm{and}\quad\iota_{B}\colon[l_{B}]\longrightarrow B$

be the order preserving morphisms. Then for $i\leq l_{A}$ and $j\leq l_{B}$,
we define $X_{(i,j)}$ to be the space resulting from having blown up
$X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ along the pullback of $Y_{1}$ and
the vanishing of $t_{\iota_{A}(k)}$ for all $k\in[i]$, and the pullback of
$Y_{2}$ and the vanishing of $t_{\iota_{B}(k)}$ for all $k\in[j]$. For now, we
may assume that the blow-ups were made in that order, though by the second
point of Proposition 3.2 this order does not actually affect the outcome. Let

$\displaystyle\beta^{A}_{(i,j)}\colon X_{(i,j)}\longrightarrow X_{(i-1,j)},$
$\displaystyle\beta^{B}_{(i,j)}\colon X_{(i,j)}\longrightarrow X_{(i,j-1)},$

denote the morphisms corresponding to each individual blow-up. We therefore
have the equality

$\beta^{B}_{(l_{A},l_{B})}\circ\cdots\circ\beta^{B}_{(l_{A},1)}\circ\beta^{A}_{(l_{A},0)}\circ\cdots\circ\beta^{A}_{(1,0)}=b.$

######  Proposition 3.2.

The expanded degeneration $X[A,B]\to C[A,B]$ satisfies the following
properties.

  1. 1.

The local model for $X[A,B]$ embeds into
$(X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1})\times(\mathbb{P}^{1})^{\lfloor
a\rfloor+\lceil b\rceil}$.

  2. 2.

The order of the blow-up morphisms $\beta^{A}_{(i,j)}$ and $\beta^{B}_{(i,j)}$
commutes.

  3. 3.

The morphism $X[A,B]\to C[A,B]$ is projective.

###### Proof.

The first point is immediate from the construction. The second point follows
from Proposition 3.1.5 of [Tsc23] and the third point follows from Proposition
3.1.7 of [Tsc23]. ∎

#### Setting terminology.

Here, for convenience, we recall some terminology from [Tsc23].

######  Definition 3.3.

We say that a dimension 2 component in a fibre of $X[A,B]\to C[A,B]$ is a
$\Delta_{1}$-component if it is contracted by the morphism
$\beta^{A}_{(i,l_{B})}$ for some $i\leq l_{A}$. Moreover if a
$\Delta_{1}$-component in a fibre is contracted by such a map then we say it
is expanded out in this fibre. We label by $\Delta_{1}^{(i)}$ the
$\Delta_{1}$-component resulting from the $i$-th blow-up along $Y_{1}$.
Similarly, a $\Delta_{2}$-component is a component which is contracted by a
morphism $\beta^{B}_{(l_{A},j)}$ for some $j\leq l_{B}$ and the
$\Delta_{2}$-component resulting from this $j$-th blow-up along $Y_{2}$ is
denoted $\Delta_{2}^{(j)}$.

We say that a dimension 2 component in a fibre of $X[A,B]\to C[A,B]$ is a
$\Delta$-component if it is a $\Delta_{i}$-component for some $i$. If it is
expanded out in some fibre we may alternatively refer to it as an expanded
component. We say that a $\Delta$-component is equal to a component $W$ of a
fibre of $X[A,B]$ if the projective coordinates associated to this
$\Delta$-component are proportional to the non-vanishing coordinates of $W$.

In the étale local notation, the $\Delta_{1}^{(i)}$ component is introduced by
the blow-up of the ideal $\langle x,t_{1}t_{2}\cdots t_{i}\rangle$ and the
$\Delta_{2}^{(n+2-j)}$ component is introduced by the blow-up of the ideal
$\langle y,t_{n+1}t_{n}\cdots t_{j}\rangle$.

######  Definition 3.4.

We refer to an irreducible component of a $\Delta$-component as a bubble. The
notions of two bubbles being equal and a bubble being expanded out in a
certain fibre follow directly from our previous definitions.

######  Definition 3.5.

We say that a $\Delta_{i}$-component is of pure type if it is not equal to any
$\Delta_{j}$-component for $j\neq i$. Otherwise we say it is of mixed type. We
will say a component is _pure of type $i$_ to say it is a
$\Delta_{i}$-component of pure type. Similarly we will refer to edges in
$\operatorname{trop}(X_{0})$ parallel to $Y_{2}\cap Y_{3}$ as being of _type
1_ and edges parallel to $Y_{1}\cap Y_{3}$ as being of _type 2_.

A $\Delta$-component in a fibre of $X[A,B]$ is said to have
$\Delta_{i}$-multiplicity $l$ if this component is equal to
$\Delta_{i}^{(j)}=\dots=\Delta_{i}^{(j+l)}$ for some $j$ and it is not equal
to any other $\Delta_{i}$-components.

Note that, in the setting of the construction $X[n]$ of [Tsc23], if both the
$\Delta_{1}$\- and $\Delta_{2}$-multiplicities of a component are nonzero,
then they must be identical.

######  Definition 3.6.

We say that a fibre in some expanded degeneration $X[A,B]$ has base
codimension $k$ if exactly $k$ basis directions vanish at this fibre. This is
independent of the value $n$.

#### Visualising the construction.

We will now describe how to visualise the fibres of $X[A,B]\to C[A,B]$. In the
following, let $(t_{1},\dots,t_{n+1})\in C[A,B]$ and let us assume that
$t_{i}$ and $t_{j}$ are two consecutive zero entries of
$(t_{1},\dots,t_{n+1})$, i.e. $t_{i}=t_{j}=0$ and $t_{i+1},\dots,t_{j-1}\neq
0$.

_Expanded components of pure type._ If both $i$ and $j$ are in $A$ and
$j\notin B$, then the component $\Delta_{1}^{(i)}=\dots=\Delta_{1}^{(j-1)}$ is
expanded out in the fibre of $X[A,B]$ over $(t_{1},\dots,t_{n+1})$ and it is
of pure type. If all other basis directions are nonzero in this fibre, then
$Y_{1}=\Delta_{1}^{(j)}=\dots=\Delta_{1}^{(\lfloor a\rfloor)}$ and $Y_{2}\cup
Y_{3}=\Delta_{1}^{(i-1)}=\dots=\Delta_{1}^{(1)}$. There is no component
$\Delta_{2}^{n+2-k}$ in the whole of $X[A,B]$ for $k\leq j$ and all
$\Delta_{2}$-components in this fibre are equal to $Y_{1}\cup Y_{3}$. Figure 2
shows such a fibre. On the right hand side of the figure, we see how to
represent this fibre tropically, where the added red vertices are the two
bubbles forming the expanded out $\Delta_{1}$-component and the edge between
them is the intersection of these bubbles. Similarly, if both $i$ and $j$ are
in $B$ and $i\notin A$, then the component
$\Delta_{2}^{(n+1-i)}=\dots=\Delta_{2}^{(n+2-j)}$ is expanded out in the fibre
of $X[A,B]$ over $(t_{1},\dots,t_{n+1})$ and of pure type.

$\Delta_{1}^{(i)}$$\cdots$$\Delta_{1}^{(j-1)}$$\Delta_{1}^{(i)}$$\cdots$$\Delta_{1}^{(j-1)}$$Y_{1}$$Y_{2}$$Y_{3}$

$\Delta_{1}^{(i)}$$\Delta_{1}^{(i)}$$Y_{1}$$Y_{3}$$Y_{2}$

Figure 2: Geometric and tropical picture at $t_{i}=t_{j}=0$ for $i,j\in
A\setminus B$ in $X[A,B]$.

_Expanded components of mixed type._ Let $t_{i}$ and $t_{j}$ be two
consecutive zero entries of $(t_{1},\dots,t_{n+1})$ as before. If $i\in A$ and
$j\in B$, then $\Delta_{1}^{(i)}=\Delta_{1}^{(n+2-j)}$ is expanded out in the
fibre of $X[A,B]$ over $(t_{1},\dots,t_{n+1})$. This situation is similar to
that discussed in [Tsc23], namely there is a bubble of mixed type in the
$\pi^{*}(Y_{1}\cap Y_{2})$ locus of such a fibre and a bubble of pure type in
each of the loci $\pi^{*}(Y_{1}\cap Y_{3})$ and $\pi^{*}(Y_{2}\cap Y_{3})$.
This is true regardless of whether $i$ is in $B$ or whether $j$ is in $A$,
however this information affects the $\Delta$-multiplicity of the $Y_{k}$
components.

_Fibres with both components of mixed and pure type._ It is of course also
possible to have fibres where both types of components described above appear.
For example, let $(t_{1},t_{2},t_{3})=(0,0,0)$, where $A\coloneqq\\{1,2\\}$
and $B\coloneqq\\{3\\}$. Then the fibre of $X[A,B]$ over this point has an
expanded component $\Delta_{1}^{(1)}$ of pure type, as well as expanded
components $\Delta_{1}^{(2)}$ and $\Delta_{2}^{(1)}$ meeting in a bubble of
mixed type in the $\pi*(Y_{1}\cap Y_{2})$ locus and separating out into two
bubbles of pure type in the $\pi*(Y_{1}\cap Y_{3})$ and $\pi*(Y_{2}\cap
Y_{3})$ loci. This is depicted in Figure 3.

$\Delta_{1}^{(2)}$$\Delta_{1}^{(1)}$$\Delta_{2}^{(1)}$$\Delta_{1}^{(1)}$$\Delta_{1}^{(2)}=\Delta_{2}^{(1)}$$Y_{1}$$Y_{2}$$Y_{3}$$\Delta_{1}^{(1)}=\Delta_{2}^{(1)}$

$\Delta_{1}^{(1)}$$\Delta_{1}^{(2)}$$\Delta_{1}^{(2)}=\Delta_{2}^{(1)}$$\Delta_{1}^{(1)}$$\Delta_{1}^{(1)}=\Delta_{2}^{(1)}$$\Delta_{2}^{(1)}$$Y_{1}$$Y_{3}$$Y_{2}$

Figure 3: Geometric and tropical picture at $t_{1}=t_{2}=t_{3}=0$ in
$X[\\{1,2\\},\\{3\\}]$.

_$\Delta_{i}$ -multiplicity of the components._ Again, let $t_{i}$ and $t_{j}$
be two consecutive zero entries of $(t_{1},\dots,t_{n+1})$. If $i\in A$, the
$\Delta_{1}$-multiplicity of the expanded component will be equal to the
number of elements of the set $\\{i,\dots,j-1\\}$ which are contained in $A$.
Recall that if $i\notin A$, then its $\Delta_{1}$-multiplicity is zero since
it is pure of type 2. Similarly, if $j\in B$, then the
$\Delta_{2}$-multiplicity of this component is given by the number of elements
of the set $\\{i+1,\dots,j\\}$ contained in $B$.

######  Remark 3.7.

The condition in Definition 3.1 is there to ensure that all modified special
fibres have the expected base codimension (see Definition 3.6). Indeed, if we
allowed, for example, $i\in B\setminus A$ and $j\in A\setminus B$ with $i<j$
and the blow-ups as given above, then when $t_{i}=t_{j}=0$ and all other basis
directions are non-vanishing, we would see a copy of $X_{0}$. Then we would
need to identify such a fibre with other copies of $X_{0}$ in our stack
$\mathfrak{X}^{\prime}$, but this would imply identifying fibres of different
base codimension, which cannot be done through isomorphisms on the base. To
avoid this unpleasantness, we therefore fix an ordering of the elements of
$A\setminus B$, $A\cap B$ and $B\setminus A$.

######  Remark 3.8.

Any modified special fibre in $X[A,B]$ can be expressed by adding certain
edges and vertices to the triangle $\operatorname{trop}(X_{0})$. Moreover, we
notice that the sequence of blow-ups which yields this modification of $X_{0}$
is completely encoded in the $Y_{1}\cap Y_{2}$ edge of this triangle. Indeed,
for each integral point on this edge, it suffices to note whether it has an
edge of type 1 or 2 attached to it, and this information completely determines
the modification.

The unbroken condition states that, in order from $Y_{1}$ to $Y_{2}$, we
should see first integral points of $Y_{1}\cap Y_{2}$ with only edges of type
2 attached, then points with both types of edges attached and, finally, on the
side closest to $Y_{2}$, we should see points with only edges of type 1
attached to them.

#### Viewing $X[A,B]$ as a sublocus of a larger scheme
$X[A^{\prime},B^{\prime}]$.

Let $n^{\prime}>n$, let $(A^{\prime},B^{\prime},n^{\prime})$ be an unbroken
pair, and suppose that $|A|\leq|A^{\prime}|$ and $|B|\leq|B^{\prime}|$, where
$|A\setminus B|=i$, $|A\cap B|=j-i$ and $|B\setminus A|=n-j$. In this case, we
may define a natural inclusion

$\iota\colon C[A,B]\longhookrightarrow C[A^{\prime},B^{\prime}],$ (3.1.2)

given by

${(t_{1},\dots,t_{i},t_{i+1},\dots,t_{j},t_{j+1}\dots,t_{n})}$${(t_{1},\dots,t_{i},1,\dots,1,t_{i+1},\dots,t_{j},1,\dots,1,t_{j+1}\dots,t_{n},1,\dots,1).}$

This, in turn, determines an embedding

$X[A,B]\longhookrightarrow X[A^{\prime},B^{\prime}].$

######  Proposition 3.9.

Given any two unbroken pairs $(A_{1},B_{1},n_{1})$ and $(A_{2},B_{2},n_{2})$
there exists a common refinement $(A,B,n)$ such that both $X[A_{1},B_{1}]$ and
$X[A_{2},B_{2}]$ embed into $X[A,B]$.

###### Proof.

We start by picking any integer $l_{A\setminus B}$ such that $|A_{1}\setminus
B_{1}|,|A_{2}\setminus B_{2}|\leq l_{A\setminus B}$. We then choose any
integer $l_{A\cap B}$ such that $|A_{1}\cap B_{1}|,|A_{2}\cap B_{2}|\leq
l_{A\cap B}$. Finally, we take any integer $l_{B\setminus A}$ such that
$|B_{1}\setminus A_{1}|,|B_{2}\setminus A_{2}|\leq l_{B\setminus A}$. Now,
recalling the above notation $[k]\coloneqq\\{1,\dots,k\\}$, let

$A\coloneqq[l_{A\setminus B}+l_{A\cap B}]\quad\textrm{and}\quad
B\coloneqq[l_{A\cap B}+l_{B\setminus A}].$

Then the embeddings (3.1.2) on the base described above induce the desired
embeddings.

∎

###  3.2 The group action and linearisation

#### The group action.

As the constructions here are generalisations of the construction $X[n]$ in
[Tsc23], we may define a group action on them in a similar way. We recall the
details here for convenience.

There is a global action of the maximal diagonal torus
$G\subset\operatorname{SL}(n+1)$ on $X[A,B]$, which we may describe as
follows. There is an isomorphism $\mathbb{G}_{m}^{n}\cong
G\subset\mathbb{G}_{m}^{n+1}$, allowing us to regard an element of $G$ as an
$(n+1)$-tuple $(\sigma_{1},\dots,\sigma_{n+1})$ where $\prod_{i}\sigma_{i}=1$.
This acts naturally on $\mathbb{A}^{n+1}$, which induces an action on $C[n]$.
The isomorphism $\mathbb{G}_{m}^{n}\cong G$ is given by

$(\tau_{1},\ldots,\tau_{n})\longrightarrow(\tau_{1},\tau_{1}^{-1}\tau_{2},\ldots,\tau_{n-1}^{-1}\tau_{n},\tau_{n}^{-1}).$

As in [Tsc23], we use the notation $(\tau_{1},\ldots,\tau_{n})$ to describe
elements of $G$ throughout this work. Now, if a $\Delta_{1}^{(k)}$ component
exists in $X[A,B]$, we denote its local $\mathbb{P}^{1}$ coordinates by
$(x_{0}^{(k)}:x_{1}^{(k)})$. Similarly, if a $\Delta_{2}^{(k)}$ component
exists in $X[A,B]$, we denote its local $\mathbb{P}^{1}$ coordinates by
$(y_{0}^{(k)}:y_{1}^{(k)})$. The following proposition is immediate from
[Tsc23].

######  Proposition 3.10.

Let $(A,B,n)$ be an unbroken pair and let $l_{A}\coloneqq|A|$ and
$l_{B}\coloneqq|B|$. We have the following properties.

  1. 1.

There is a unique $G$-action on $X[A,B]$ such that $X[A,B]\to
X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ is equivariant with respect to the
natural action of $G$ on $\mathbb{A}^{n+1}$.

  2. 2.

In the étale local model, this action is the restriction of the action on
$(X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1})\times(\mathbb{P}^{1})^{l_{A}+l_{B}}$,
which is trivial on $X$, acts by

$\displaystyle t_{1}$ $\displaystyle\longmapsto\tau_{1}^{-1}t_{1}$
$\displaystyle t_{k}$ $\displaystyle\longmapsto\tau_{k}^{-1}\tau_{k-1}t_{k}$
$\displaystyle t_{n+1}$ $\displaystyle\longmapsto\tau_{n}t_{n+1}$

on the basis directions, and by

$\displaystyle(x_{0}^{(k)}:x_{1}^{(k)})$
$\displaystyle\longmapsto(\tau_{k}x_{0}^{(k)}:x_{1}^{(k)})$
$\displaystyle(y_{0}^{(k)}:y_{1}^{(k)})$
$\displaystyle\longmapsto(y_{0}^{(k)}:\tau_{n+1-k}y_{1}^{(k)})$

on the components $\Delta_{1}^{(k)}$ and $\Delta_{2}^{(k)}$ which exist in
$X[A,B]$.

  3. 3.

The embeddings $X[A,B]\hookrightarrow X[A^{\prime},B^{\prime}]$ given in
(3.1.2) are equivariant under the group action.

#### The $G$-linearisation.

Similarly to [Tsc23], we may embed $X[A,B]$ into a product of projective
bundles. This is done again by defining vector bundles

$\displaystyle\mathcal{F}_{1}^{(i)}$
$\displaystyle=\operatorname{pr}_{1}^{*}\mathcal{O}_{X}(-Y_{1})\oplus\operatorname{pr}_{2}^{*}\mathcal{O}_{\mathbb{A}^{n+1}}(-V(t_{i}))$
$\displaystyle\mathcal{F}_{2}^{(j)}$
$\displaystyle=\operatorname{pr}_{1}^{*}\mathcal{O}_{X}(-Y_{2})\oplus\operatorname{pr}_{2}^{*}\mathcal{O}_{\mathbb{A}^{n+1}}(-V(t_{n+2-j}))$

on $X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ for $i\in A$ and $j\in B$, where
$\operatorname{pr}_{i}$ denote the natural projections from
$X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$ to $X$ and $\mathbb{A}^{n+1}$. Then
the following lemma follows directly from Lemma 3.2.1 of [Tsc23].

######  Lemma 3.11.

There is an embedding

$X[A,B]\longhookrightarrow\prod_{i,j}\mathbb{P}(\mathcal{F}_{i}^{(j)}),$

where the product of projective bundles
$\prod_{i,j}\mathbb{P}(\mathcal{F}_{i}^{(j)})$ is constructed as a fibre
product over $X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$. Etale locally this
corresponds to the embedding in
$(X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1})\times(\mathbb{P}^{1})^{l_{A}+l_{B}}$.

We may then construct the following $G$-linearised ample line bundles on
$X[A,B]$.

######  Lemma 3.12.

There exists a $G$-linearised ample line bundle $\mathcal{L}$ on $X[A,B]$ such
that locally the lifts to this line bundle of the $G$-action on each
$\mathbb{P}^{1}$ corresponding to a $\Delta_{1}^{(i)}$ and on each
$\mathbb{P}^{1}$ corresponding to a $\Delta_{2}^{(n+2-j)}$ are given by

$\displaystyle(x_{0}^{(i)};x_{1}^{(i)})$
$\displaystyle\longmapsto(\tau_{i}^{a_{i}}x_{0}^{(i)};\tau_{i}^{-b_{i}}x_{1}^{(i)})$
(3.2.1) $\displaystyle(y_{0}^{(n+2-j)};y_{1}^{(n+2-j)})$
$\displaystyle\longmapsto(\tau_{j+1}^{-c_{j}}y_{0}^{(n+2-j)};\tau_{j+1}^{d_{j}}y_{1}^{(n+2-j)})$
(3.2.2)

for any $i\in A$ and $j\in B$ and any choice of positive integers
$a_{i},b_{i},c_{j},d_{j}$.

###### Proof.

This follows directly from Lemma 3.2.2 of [Tsc23]. ∎

As before, we will apply the relative Hilbert-Mumford criterion of [GHH15] to
define GIT stability conditions with respect to the $G$-linearised line
bundles we defined. We leave this discussion for Section 4.3.

##  4 The stack construction

Building upon the scheme constructions $X[A,B]\to C[A,B]$, we define a stack
of expansions and its family, which we denote
$\mathfrak{X}^{\prime}\to\mathfrak{C}^{\prime}$. The stacks $\mathfrak{X}$ and
$\mathfrak{C}$ of [Tsc23] are substacks of $\mathfrak{X}^{\prime}$ and
$\mathfrak{C}^{\prime}$; the latter contains more choices of expansions of
$X$. Note that it does not contain all possible choices, as we have made
restrictions for geometric reasons on the type of expansions we allow. For
example, in order to construct a large family containing all the expansions,
we chose them so that the order of the blow-ups commutes.

###  4.1 The stack of expansions

In the following we define the stack of expansions $\mathfrak{C}^{\prime}$.
This will differ slightly from [Tsc23] and the construction of Li and Wu in
that here we want to remember what components of the special fibre $X_{0}$ are
blown up along which basis directions, i.e. the data of the sets $A$ and $B$.
We will start by defining isomorphisms which when lifted to the family will
effectively identify any two isomorphic fibres.

#### The isomorphisms.

Let us consider $\mathbb{A}^{n+1}$ with its natural torus action
$\mathbb{G}_{m}^{n}$ as before and denote by $(\mathbb{A}^{n+1})_{[A,B]}$ this
space enhanced by a choice of sets $A$ and $B$ such that $(A,B,n)$ is
unbroken. We label elements of $(\mathbb{A}^{n+1})_{[A,B]}$ by
$(t_{1},\dots,t_{n+1})$ as before and, for any $k$, we use again the notation
$[k]\coloneqq\\{1,\dots,k\\}$. Let $J\subseteq[n+1]$ and let
$J^{\circ}=[n+1]\setminus J$ be its complement. We denote the cardinality of
$J$ by $r\coloneqq|J|$ and define

$\displaystyle\operatorname{ind}_{J}\colon[r]\longrightarrow
J\quad\textrm{and}$
$\displaystyle\operatorname{ind}_{J^{\circ}}\colon\\{r+1,\dots,n+1\\}\longrightarrow
J^{\circ}$

to be the order preserving isomorphisms.

Remark that, although the map $\operatorname{ind}_{J}$ is defined identically
to the map of the same name in Section 5 of [Tsc23], the map
$\operatorname{ind}_{J^{\circ}}$ is given a different definition here. This
change is introduced so that we may keep track of which indices of the entries
of $(t_{1},\dots,t_{n+1})$ lie in $A$ or $B$ in the following isomorphisms.

Let

$(\mathbb{A}^{n+1}_{J})_{[A,B]}=\\{(t)\in(\mathbb{A}^{n+1})_{[A,B]}|\
t_{i}=0,\ i\in J\\}\subset(\mathbb{A}^{n+1})_{[A,B]}$

and

$(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}=\\{(t)\in(\mathbb{A}^{n+1})_{[A,B]}|\
t_{i}\neq 0,\ i\in J^{\circ}\\}\subset(\mathbb{A}^{n+1})_{[A,B]}.$

Now, given $(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}$, we define a corresponding
space
$(\mathbb{A}^{r}\times\mathbb{G}_{m}^{n+1-r})_{[\epsilon(A),\epsilon(B)]}$ by
specifying sets $\epsilon(A)$ and $\epsilon(B)$ in the following way. If
$\operatorname{ind}_{J}(1)\in A\cap B$, then let
$1\in\epsilon(A)\setminus\epsilon(B)$. Then, for $i>1$, if
$\operatorname{ind}_{J}(i)\in A\cap B$ then $i\in\epsilon(A)\cap\epsilon(B)$.
For $i\geq 1$, if $\operatorname{ind}_{J}(i)\in A\setminus B$ then
$i\in\epsilon(A)\setminus\epsilon(B)$ and if $\operatorname{ind}_{J}(i)\in
B\setminus A$ then $i\in\epsilon(B)\setminus\epsilon(A)$. Finally, let
$i\in\epsilon(B)\setminus\epsilon(A)$ for all $i>r$. We note that
$(\epsilon(A),\epsilon(B),n)$ is unbroken. We may then define the isomorphism

$\tau_{J}^{[A,B]}\colon(\mathbb{A}^{r}\times\mathbb{G}_{m}^{n+1-r})_{[\epsilon(A),\epsilon(B)]}\longrightarrow(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}$

given by

$(a_{1},\ldots,a_{r},\sigma_{r+1},\dots,\sigma_{n+1})\longrightarrow(t_{1},\dots,t_{n+1}),$

where

$\displaystyle t_{i}=a_{j},\ \mathrm{if}\ \operatorname{ind}_{J}(j)=i,$
$\displaystyle t_{i}=\sigma_{j},\ \mathrm{if}\
\operatorname{ind}_{J^{\circ}}(j)=i.$

Essentially, here, we have rearranged the order of the basis directions, so
that those which can vanish are given by the $a_{i}$ entries at the front of
the basis vector. These entries retain their ordering relative to each other
and their partition with respect to the unbroken pair, except that the index
of the first entry may go from $A\cap B$ to $\epsilon(A)\setminus\epsilon(B)$.
The $\sigma_{i}$ entries correspond to the nonzero $t_{i}$ and their indices
are all contained in $\epsilon(B)\setminus\epsilon(A)$.

Given two triples $(A,B,J)$ and $(A^{\prime},B^{\prime},J^{\prime})$ such that
$|J|=|J^{\prime}|$ and such that $\epsilon(A)=\epsilon(A^{\prime})$ and
$\epsilon(B)=\epsilon(B^{\prime})$, we may then define an isomorphism

$\tau_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}=\tau_{J}^{[A,B]}\circ(\tau_{J^{\prime}}^{[A^{\prime},B^{\prime}]})^{-1}\colon(\mathbb{A}^{n+1}_{U(J^{\prime})})_{[A^{\prime},B^{\prime}]}\longrightarrow(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}.$

Now, we define a second set of isomorphisms similarly to the above. Given any
$(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}$, we may define an alternative space
$(\mathbb{A}^{r}\times\mathbb{G}_{m}^{n+1-r})_{[\delta(A),\delta(B)]}$ by
describing sets $\delta(A)$ and $\delta(B)$ in the following way. If
$\operatorname{ind}_{J}(r)\in A\cap B$, then let
$r\in\delta(B)\setminus\delta(A)$. For $i<r$, if $\operatorname{ind}_{J}(i)\in
A\cap B$, then let $i\in\delta(A)\cap\delta(B)$. For $i\leq r$, if
$\operatorname{ind}_{J}(i)\in A\setminus B$, then let
$i\in\delta(A)\setminus\delta(B)$ and if $\operatorname{ind}_{J}(i)\in
B\setminus A$ let $i\in\delta(B)\setminus\delta(A)$. Again, the pair
$(\delta(A),\delta(B),n)$ is unbroken. We then define an isomorphism

$\rho_{J}^{[A,B]}\colon(\mathbb{A}^{r}\times\mathbb{G}_{m}^{n+1-r})_{[\delta(A),\delta(B)]}\longrightarrow(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}$

given by

$(a_{1},\ldots,a_{r},\sigma_{r+1},\dots,\sigma_{n+1})\longrightarrow(t_{1},\dots,t_{n+1}),$

where

$\displaystyle t_{i}=a_{j},\ \mathrm{if}\ \operatorname{ind}_{J}(j)=i,$
$\displaystyle t_{i}=\sigma_{j},\ \mathrm{if}\
\operatorname{ind}_{J^{\circ}}(j)=i.$

As above, given two triples $(A,B,J)$ and $(A^{\prime},B^{\prime},J^{\prime})$
such that $|J|=|J^{\prime}|$ and such that $\delta(A)=\delta(A^{\prime})$ and
$\delta(B)=\delta(B^{\prime})$, we may then specify an isomorphism

$\rho_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}=\rho_{J}^{[A,B]}\circ(\rho_{J^{\prime}}^{[A^{\prime},B^{\prime}]})^{-1}\colon(\mathbb{A}^{n+1}_{U(J^{\prime})})_{[A^{\prime},B^{\prime}]}\longrightarrow(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}.$

#### The stack $\mathfrak{C}^{\prime}$.

Finally, given an unbroken pair $(A,B,n)$, we define $\mathfrak{U}^{A,B}$ to
be the quotient $[(\mathbb{A}^{n+1})_{[A,B]}/\\!\\!\sim]$ by the equivalences
generated by the $\mathbb{G}_{m}^{n}$-action and the equivalences
$\tau_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$ and
$\rho_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$ for compatible triples
$(A,B,J)$ and $(A^{\prime},B^{\prime},J^{\prime})$. Recall from Section 3.1
that we had natural inclusions (3.1.2)

$C[A,B]\longhookrightarrow C[A^{\prime\prime},B^{\prime\prime}],$

for all $A^{\prime\prime}$ and $B^{\prime\prime}$ such that
$(A^{\prime\prime},B^{\prime\prime},n^{\prime\prime})$ is unbroken for some
$n^{\prime\prime}>n$ and such that $|A|\leq|A^{\prime\prime}|$ and
$|B|\leq|B^{\prime\prime}|$. These induce open immersions of stacks

$\displaystyle\mathfrak{U}^{A,B}\longrightarrow\mathfrak{U}^{A^{\prime\prime},B^{\prime\prime}}.$

Let $\mathfrak{U}^{\prime}\coloneqq\lim\limits_{\to}\mathfrak{U}^{A,B}$ be the
direct limit over all unbroken pairs $(A,B,n)$ and let
$\mathfrak{C}^{\prime}\coloneqq
C\times_{\mathbb{A}^{1}}\mathfrak{U}^{\prime}$.

We now make a few remarks concerning the isomorphisms described in this
section. As mentioned above, these isomorphisms are effectively a reordering
of the basis directions, where the $t_{i}$’s whose indices lie in $J$ preserve
their order relative to each other and the corresponding indices preserve
their $A,B$ labelling, except in certain cases for the first and last index in
$J$. As we will see in the proof of Proposition 4.1, these are exactly the
isomorphisms on the base which correspond to isomorphisms of the corresponding
fibres above these elements.

Let $(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}$ be such that
$\operatorname{ind}_{J}(1)\in A\cap B$. Then it is isomorphic by the
isomorphism $\tau_{J}^{[A,B]}$ to the space
$(\mathbb{A}^{r}\times\mathbb{G}_{m}^{n+1-r})_{[\epsilon(A),\epsilon(B)]}$
defined above, where $1\notin\epsilon(B)$. We note that the isomorphisms
$\tau_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$ identify
$(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}$ with spaces
$(\mathbb{A}^{n+1}_{U(J^{\prime})})_{[A^{\prime},B^{\prime}]}$ where either
$\operatorname{ind}_{J^{\prime}}(1)\in A^{\prime}\cap B^{\prime}$ or
$\operatorname{ind}_{J^{\prime}}(1)\in A^{\prime}\setminus B^{\prime}$, as
long as $|J|=|J^{\prime}|$ and the equalities
$\epsilon(A)=\epsilon(A^{\prime})$ and $\epsilon(B)=\epsilon(B^{\prime})$
hold.

###  4.2 The family $\mathfrak{X}^{\prime}$ over $\mathfrak{C}^{\prime}$.

Let $X[A,B]\to C[A,B]$ be a construction as described in Section 3.1 and
recall that $\pi\colon X[A,B]\to X$ is the projection to the original family.
Let $n^{\prime}>n$, let $(A^{\prime},B^{\prime},n^{\prime})$ be unbroken, and
let $|A|\leq|A^{\prime}|$ and $|B|\leq|B^{\prime}|$. In such a case, we
defined a natural inclusion

$\iota\colon C[A,B]\longhookrightarrow C[A^{\prime},B^{\prime}]$

in (3.1.2). Then the induced family
$(\iota^{*}X[A^{\prime},B^{\prime}],\iota^{*}\pi)$ is isomorphic to
$(X[A,B],\pi)$ over $C[A,B]$. The following proposition shows that the
equivalences defined on the stack $\mathfrak{U}^{A,B}$ lift to
$C$-isomorphisms of fibres. Let $X[A,B]_{U(J)}$ and $X[A,B]_{J}$ be the
restrictions of $X[A,B]$ to $(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}$ and
$(\mathbb{A}^{n+1}_{J})_{[A,B]}$ respectively.

######  Proposition 4.1.

Let $(A,B,n)$ and $(A^{\prime},B^{\prime},n)$ define two unbroken pairs. The
schemes $X[A,B]_{U(J)}$ and $X[A^{\prime},B^{\prime}]_{U(J^{\prime})}$ are
isomorphic if and only if $(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}$ and
$(\mathbb{A}^{n+1}_{U(J^{\prime})})_{[A^{\prime},B^{\prime}]}$ are related by
the isomorphisms $\tau_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$ and
$\rho_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$.

###### Proof.

It is clear that $X[A,B]_{U(J)}$ and
$X[A^{\prime},B^{\prime}]_{U(J^{\prime})}$ are isomorphic on
$\pi^{-1}_{[A,B]}(X^{\circ})\cong\pi^{-1}_{[A^{\prime},B^{\prime}]}(X^{\circ})$,
i.e. on the locus where all basis directions are nonzero. We must show that
for each modified special fibre in $X[A,B]_{U(J)}$ there is an isomorphic
fibre in $X[A^{\prime},B^{\prime}]_{U(J^{\prime})}$ if and only if the
corresponding bases are related by
$\tau_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$ and
$\rho_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$. Then the proposition
follows from the fact that if there is a bijection between the isomorphism
classes of fibres of $X[A,B]$ and those of $X[A^{\prime},B^{\prime}]$, the two
spaces must be isomorphic as schemes, as they must both be given by the same
sequence of blow-ups on $X\times_{\mathbb{A}^{1}}\mathbb{A}^{n+1}$. As the
blow-ups commute, we may discover this sequence just by looking at the fibres
where all entries of $(t_{1},\dots,t_{n+1})$ are zero (in fact, by
construction, it is enough to show that these two fibres of maximal base
codimension are isomorphic to show that the schemes $X[A,B]$ and
$X[A^{\prime},B^{\prime}]$ are isomorphic).

First, assume there is a sequence of
$\tau_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$ and
$\rho_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$ isomorphisms such that
$(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}\cong(\mathbb{A}^{n+1}_{U(J^{\prime})})_{[A^{\prime},B^{\prime}]}$.
Since $|J|=|J^{\prime}|=r$, this implies that, for $1<i<r$, the element
$\operatorname{ind}_{J}(i)$ belongs to $A\setminus B$ if and only if
$\operatorname{ind}_{J^{\prime}}(i)$ belongs to $A^{\prime}\setminus
B^{\prime}$. The same goes for the sets $A\cap B$ with $A^{\prime}\cap
B^{\prime}$ and $B\setminus A$ with $B^{\prime}\setminus A^{\prime}$.

Next, we note that the placement of the nonzero entries in
$(t_{1},\dots,t_{n+1})$ influences the $\Delta$-multiplicity of the components
but has no effect on how many components are expanded out in the fibre above
this point or whether these expanded components are of type $\Delta_{1}$ or
type $\Delta_{2}$. Moreover, changing the first zero entry of
$(t_{1},\dots,t_{n+1})$ from an element of $A\setminus B$ to an element of
$A\cap B$ or vice versa also does not affect which components are expanded out
in this fibre. The same is true of changing the last zero entry of
$(t_{1},\dots,t_{n+1})$ from $B\setminus A$ to $A\cap B$ or vice versa. This
can be easily seen by studying the equations of the blow-ups. It therefore
must follow that a sequence of isomorphisms
$\tau_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$ and
$\rho_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$, as they only modify the
base in ways that preserve the fibres, induces an isomorphism
$X[A,B]_{U(J)}\cong X[A^{\prime},B^{\prime}]_{U(J^{\prime})}$.

Conversely, let us assume that all fibres in $X[A,B]_{U(J)}$ are isomorphic to
fibres in $X[A^{\prime},B^{\prime}]_{U(J^{\prime})}$. This means that for
every fibre of $X[A,B]_{U(J)}$ over a point $(t_{1},\dots,t_{n+1})$ of the
base $(\mathbb{A}^{n+1}_{U(J)})_{[A,B]}$, there exists a fibre of
$X[A^{\prime},B^{\prime}]_{U(J^{\prime})}$ over a point
$(t^{\prime}_{1},\dots,t^{\prime}_{n+1})$ of the base
$(\mathbb{A}^{n+1}_{U(J^{\prime})})_{[A^{\prime},B^{\prime}]}$ such that the
exact same number of components are expanded out in the same loci of both
fibres and these components must have the same type. Clearly, there must be
the same amount of zero entries in $(t_{1},\dots,t_{n+1})$ and
$(t^{\prime}_{1},\dots,t^{\prime}_{n+1})$; call this number $k$. Moreover, for
$k>1$, by studying the equations of the blow-ups we can see that for the above
statement to be true, the first $k-1$ zeros of $(t_{1},\dots,t_{n+1})$ must
have indices belonging to $A$ if and only if the first $k-1$ zeros of
$(t^{\prime}_{1},\dots,t^{\prime}_{n+1})$ have indices belonging to
$A^{\prime}$; and the last $k-1$ zeros of $(t_{1},\dots,t_{n+1})$ must have
indices belonging to $B$ if and only if the last $k-1$ zeros of
$(t^{\prime}_{1},\dots,t^{\prime}_{n+1})$ have indices belonging to
$B^{\prime}$.

If the index of the first zero in $(t_{1},\dots,t_{n+1})$ is not contained in
$A$, then it must be contained in $B\setminus A$ and all expanded components
in the fibre above this point are of type $\Delta_{2}$. This can only be
isomorphic to the fibre above the point
$(t^{\prime}_{1},\dots,t^{\prime}_{n+1})$ if the indices of all zero entries
in $(t^{\prime}_{1},\dots,t^{\prime}_{n+1})$ are contained in
$B^{\prime}\setminus A^{\prime}$. If the index of the first zero in
$(t_{1},\dots,t_{n+1})$ is contained in $A$ and the indices of all subsequent
zeros are contained in $B$, however, then whether or not this first index is
contained in $B$ has no effect on the components expanded out in this fibre.
In conclusion, if the fibres above points $(t_{1},\dots,t_{n+1})$ and
$(t^{\prime}_{1},\dots,t^{\prime}_{n+1})$ are isomorphic, then the index of
the first zero entry of $(t_{1},\dots,t_{n+1})$ is in $B\setminus A$ if and
only if the index of the first zero entry of
$(t^{\prime}_{1},\dots,t^{\prime}_{n+1})$ is in $B^{\prime}\setminus
A^{\prime}$; and if the index of the first zero entry of
$(t_{1},\dots,t_{n+1})$ is in $A\cap B$ then the index of the first zero entry
of $(t^{\prime}_{1},\dots,t^{\prime}_{n+1})$ is in either of
$A^{\prime}\setminus B^{\prime}$ or $A^{\prime}\cap B^{\prime}$. A similar
reasoning holds with respect to the index of the last zero entry being
contained in $B\setminus A$ or $A\cap B$. Indeed, if this index is contained
in $A\cap B$, then $\operatorname{ind}_{J^{\prime}}(k)$ may be contained in
$A^{\prime}\cap B^{\prime}$ or $B^{\prime}\setminus A^{\prime}$. But these are
exactly the elements of the base which are related by the isomorphisms
$\tau_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$ and
$\rho_{J,[A,B]}^{J^{\prime},[A^{\prime},B^{\prime}]}$. ∎

We define $\mathfrak{X}^{A,B}$ to be the quotient $[X[A,B]/\\!\\!\sim]$ by the
equivalences generated by the $\mathbb{G}_{m}^{n}$-action and equivalences
lifted from $\mathfrak{U}^{A,B}$. There are natural immersions of stacks

$\displaystyle\mathfrak{X}^{A,B}\longhookrightarrow\mathfrak{X}^{A^{\prime},B^{\prime}},$

induced by the immersions
$\mathfrak{U}^{A,B}\hookrightarrow\mathfrak{U}^{A^{\prime},B^{\prime}}$.
Finally, we define $\mathfrak{X}=\lim\limits_{\to}\mathfrak{X}^{A,B}$ to be
the direct limit over all unbroken pairs $(A,B,n)$. It is an Artin stack (here
we mean as opposed to a Deligne-Mumford stack, i.e. it has infinite
automorphisms).

###  4.3 Stability conditions

We now discuss stability conditions on the stack $\mathfrak{X}^{\prime}$. We
generalise the two perspectives given in [Tsc23] to this case, namely that
coming from the GIT stability conditions arising from the $G$-linearised line
bundles we constructed in Section 3.1, and that which relates to the stability
conditions presented by Li and Wu in [LW15] and Maulik and Ranganathan in
[MR24].

#### GIT stability on the relative Hilbert scheme of points.

Let $(A,B,n)$ be an unbroken pair and denote by

$H^{m}_{[A,B]}\coloneqq\operatorname{Hilb}^{m}(X[A,B]/C[A,B])$

the relative Hilbert scheme of $m$ points. Recall that in Lemma 3.12, we gave
possible choices for a $G$-linearised ample line bundle $\mathcal{L}$ on
$X[A,B]$. As in [Tsc23], this will induce a $G$-linearised ample line bundle
on $H^{m}_{[A,B]}$ in the following way. Let

$Z^{m}_{[A,B]}\subset H^{m}_{[A,B]}\times_{C[A,B]}X[A,B]$

be the universal family, with first and second projections $p$ and $q$. The
line bundle

$\mathcal{M}_{l}\coloneqq\det p_{*}(q^{*}\mathcal{L}^{\otimes
l}|_{Z^{m}_{[A,B]}})$

is relatively ample when $l\gg 0$ and is $G$-linearised, by the same argument
as in [GHH19]. The choices of line bundle $\mathcal{M}_{l}$ on $H^{m}_{[A,B]}$
arising in this way paired with choices of lifts of the group action to
$\mathcal{M}_{l}$ are enough to allow us to construct the desired stable
locus. The following results follow directly from Theorem 4.4.3 and Lemma
4.4.1 of [Tsc23].

######  Theorem 4.2.

Let $(A,B,n)$ be an unbroken pair and let $Z$ be a length $m$ zero-dimensional
subscheme in a fibre of $X[A,B]$. Then there exists a GIT stability condition
on $H^{m}_{[A,B]}$ which makes $Z$ stable (as opposed to strictly semistable)
if and only if there is at least one point of the support of $Z$ in the union
$(\Delta_{1}^{(k)})^{\circ}\cup(\Delta_{2}^{(n+1-k)})^{\circ}$ for every $k$.

######  Corollary 4.3.

Theorem 4.2 still holds if we restrict the possible GIT stability conditions
to just those arising from choices of line bundles $\mathcal{M}_{l}$ as
defined above.

#### Li-Wu and modified GIT stability.

We extend the notions of stability set up in [Tsc23], namely LW and SWS
stability, first on schemes, then generalise to stacks. In section 5.2, we
will introduce an additional stability condition needed to obtain proper
algebraic stacks. Throughout this section we assume that $(A,B,n)$ is an
unbroken pair.

######  Definition 4.4.

Let $Z$ be a length $m$ zero-dimensional subscheme in a fibre of $X[A,B]$. We
say that $Z$ is _Li-Wu (LW) stable_ if $Z$ is smoothly supported and has
finite automorphisms. We denote the LW stable locus by
$H^{m}_{[A,B],\operatorname{LW}}$.

######  Remark 4.5.

In the case of Hilbert schemes of points, this happens to correspond to the
notion of stability defined in [LW15]. If we wanted to generalise these
results to Hilbert schemes with non-constant polynomials, however, we would
need to take the original definition of Li-Wu stability which can be found in
[LW15] or [Tsc23].

We also extend the definition of SWS stability to this setting in the obvious
way as follows.

######  Definition 4.6.

Let $Z$ be a length $m$ zero-dimensional subscheme in a fibre of $X[A,B]$. We
say that $Z$ is weakly strictly stable if there exists any $G$-linearised
ample line bundle $\mathcal{M}^{\prime}$ on $H^{m}_{[A,B]}$ with respect to
which $Z$ is GIT stable. The term _strictly_ is used here to emphasize that we
exclude strictly semistable points. If $Z$ is also supported in the smooth
locus of the fibre, then we say it is smoothly supported weakly strictly
stable (abbreviated to _SWS stable_ as before) and we denote the SWS stable
locus by $H^{m}_{[A,B],\operatorname{SWS}}$.

We then have the inclusion

$H^{m}_{[A,B],\operatorname{SWS}}\subset H^{m}_{[A,B],\operatorname{LW}},$

since, if points are GIT stable, they must have finite stabilisers. We now
extend our definition of SWS stability to the stack $\mathfrak{X}^{\prime}$.

Given any $C$-scheme $S$, an object of $\mathfrak{X}^{\prime}(S)$ is a
pullback family $\xi^{*}X[A,B]$ for a morphism

$\xi\colon S\to C[A,B].$

######  Definition 4.7.

Let $\mathcal{X}\in\mathfrak{X}^{\prime}(S)$ and let $Z$ be family of length
$m$ zero-dimensional subschemes in $\mathcal{X}$. The pair $(Z,\mathcal{X})$
is said to be SWS stable if and only if $\mathcal{X}\coloneqq\xi^{*}X[A,B]$
for some morphism $\xi\colon S\to C[A,B]$, where $Z$ is supported in the
smooth locus of $\mathcal{X}$ and there exists some $G$-linearised ample line
bundle on $H^{m}_{[A,B]}$ which makes $\xi_{*}Z$ be GIT stable.

LW stability can be extended to the stack setting similarly. It is important
to note that an object $\mathcal{X}\in\mathfrak{X}^{\prime}(S)$ is an
equivalence class under the action of $G$ and the isomorphisms set up in
Section 4.1 which identify isomorphic fibres. In particular, we highlight that
the above definition means that the entire equivalence class given by a pair
$(Z,\mathcal{X})$ is SWS stable if there is any representative of the
equivalence class of $\mathcal{X}$ for which this pair is SWS stable.

######  Proposition 4.8.

There exists a pair $(Z,\mathcal{X})$ for some
$\mathcal{X}\in\mathfrak{X}^{\prime}(k)$ (where $k$ is an algebraically closed
field of characteristic zero as before) which is LW stable but not SWS stable.
This means there is no representative of the equivalence class of
$\mathcal{X}$ for which this pair is SWS stable.

###### Proof.

This can be seen easily by describing an example where this happens. Take
$\mathcal{X}$ to be a fibre of base codimension 2 where exactly one
$\Delta$-component is expanded out and it is pure of type 1. Then let $Z$ be a
length $m$ zero dimensional subscheme with at least part of its support lying
in the interior of this expanded $\Delta_{1}$ component, but no point of its
support lying in $Y_{1}$. This will be LW stable but it will never be SWS
stable because no matter what representative of the equivalence class of
$\mathcal{X}$ we choose, this representative will always have at least one
$\Delta_{1}$\- and one $\Delta_{2}$-component equal to $Y_{1}$. By Theorem
4.2, the pair cannot be SWS stable. ∎

######  Remark 4.9.

If we restrict ourselves to the substack $\mathfrak{X}$, the LW and SWS
stability conditions are the same up to the equivalences of the stack. In that
context, if there is at least one point of the support of $Z$ in the union
$(\Delta_{1}^{(i)})^{\circ}\cup(\Delta_{2}^{(n+1-i)})^{\circ}$ for every $i$
for which these components are expanded out in $\mathcal{X}$, then
$(Z,\mathcal{X})$ is SWS stable. This is because in $\mathfrak{X}$ we have
restricted the type of modified fibre we can have, so that examples like the
one given in the proof of Proposition 4.8 are not included. We may easily see
that in this restricted setting, given any such pair $(Z,\mathcal{X})$, there
is an equivalent pair under the isomorphisms of the stack $\mathfrak{X}$
satisfying the condition that at least one point of the support of $Z$ lies in
the union $(\Delta_{1}^{(i)})^{\circ}\cup(\Delta_{2}^{(n+1-i)})^{\circ}$ for
every $i$. Such a pair is therefore SWS stable by Theorem 4.2.

#### Stacks of stable objects.

Let us denote by $\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$ the stacks of SWS and LW stable length
$m$ zero-dimensional subschemes in $\mathfrak{X}^{\prime}$. Let $S$ be a
$C$-scheme. An object of $\mathfrak{N}^{n}_{\operatorname{SWS}}(S)$ is defined
to be a pair $(Z,\mathcal{X})$, where $\mathcal{X}\in\mathfrak{X}^{\prime}(S)$
and $Z$ is an $S$-flat SWS stable family in $\mathcal{X}$. Similarly, an
object of $\mathfrak{N}^{n}_{\operatorname{LW}}(S)$ is a pair
$(Z,\mathcal{X})$, where $\mathcal{X}\in\mathfrak{X}^{\prime}(S)$ and $Z$ is
an $S$-flat LW stable family in $\mathcal{X}$.

######  Remark 4.10.

Unlike the stacks $\mathfrak{M}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{M}^{m}_{\operatorname{LW}}$ of [Tsc23], the stacks
$\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$ are universally closed but not
separated, meaning that stable families of length $m$ zero-dimensional
subschemes in these stacks may have more than one limit. As a consequence, Li-
Wu stability and Maulik-Ranganathan stability do not coincide on these stacks.
Indeed, it does not yet make sense here to speak of Maulik-Ranganathan
stability on the stack $\mathfrak{X}^{\prime}$ as we have not defined a notion
of tube component and a choice of stable limit among all possible limits for a
given a family $(Z,\mathcal{X})$. As we will see, in order to construct
separated stacks we will need to select one of two options. The first is to
identify all choices of representatives for a limit, which will yield a stack
which is no longer algebraic. This parallels work of Kennedy-Hunt [Ken23]. The
second is to make a choice of substack which picks out exactly one
representative for each limit and these substacks will be isomorphic to the
underlying stacks produced by the methods of Maulik and Ranganathan in [MR24]
(the word underlying here is used to signify that we do not speak of
logarithmic structures).

###  4.4 Universal closure

We will now show that the stacks $\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$ are universally closed. Part of this
proof follows easily from the fact that $\mathfrak{X}$ is a substack of
$\mathfrak{X}^{\prime}$ and universal closure results of [Tsc23]; we will,
however, give a more detailed proof of this statement here, as this will allow
us to illustrate the fact that neither stack is separated, and where these
different choices of limit representatives for a family of length $m$ zero-
dimensional subschemes occur. We start by recalling the following definition
from [Tsc23].

######  Definition 4.11.

Let $S\coloneqq\operatorname{Spec}R\to C$, where $R$ is some discrete
valuation ring and let $\eta$ denote the generic point of $S$. Now, let
$(Z,\mathcal{X})$ be a pair where $\mathcal{X}\in\mathfrak{X}(S)$ and $Z$ is
an $S$-flat family of length $m$ zero-dimensional subschemes in $\mathcal{X}$.
Let $S^{\prime}\to S$ be some finite base change and denote the generic and
closed points of $S^{\prime}$ by $\eta^{\prime}$ and $\eta^{\prime}_{0}$
respectively. We say that a pair
$(Z^{\prime}_{\eta_{0}^{\prime}},\mathcal{X}^{\prime}_{\eta_{0}^{\prime}})$ is
an extension of $(Z_{\eta},\mathcal{X}_{\eta})$ if there exists such a base
change and
$(Z^{\prime}_{\eta_{0}^{\prime}},\mathcal{X}^{\prime}_{\eta_{0}^{\prime}})$ is
the restriction to $\eta^{\prime}_{0}$ of some $S^{\prime}$-flat family
$(Z^{\prime},\mathcal{X}^{\prime})$ with
$\mathcal{X}^{\prime}\in\mathfrak{X}^{\prime}(S^{\prime})$ such that
$Z_{\eta}\times_{\eta}\eta^{\prime}\cong Z^{\prime}_{\eta^{\prime}}$ and
$\mathcal{X}_{\eta}\times_{\eta}\eta^{\prime}\cong\mathcal{X}^{\prime}_{\eta^{\prime}}$.

######  Proposition 4.12.

The stack $\mathfrak{N}^{m}_{\operatorname{SWS}}$ is universally closed.

###### Proof.

Let $S\coloneqq\operatorname{Spec}R\to C$, where $R$ is some discrete
valuation ring with uniformising parameter $w$ and quotient field $k$. We
denote by $\eta$ and $\eta_{0}$ the generic and closed points of $S$
respectively. Let $(Z,\mathcal{X})$ be an $S$-flat family of length $m$ zero-
dimensional subschemes such that $\mathcal{X}\in\mathfrak{X}^{\prime}(S)$ and
$(Z_{\eta},\mathcal{X}_{\eta})\in\mathfrak{N}^{m}_{\operatorname{SWS}}(\eta)$.
As in the proof of Proposition 6.1.3 of [Tsc23], we show that there exists a
finite base change $S^{\prime}\coloneqq\operatorname{Spec}R^{\prime}\to S$,
for some discrete valuation ring $R^{\prime}$ and a pair
$(Z^{\prime},\mathcal{X}^{\prime})\in\mathfrak{N}^{m}_{\operatorname{SWS}}(S^{\prime})$
satisfying the following condition. We denote by $\eta^{\prime}$ and
$\eta_{0}^{\prime}$ the generic and closed points of $S^{\prime}$
respectively. Then $S^{\prime}$ and $(Z^{\prime},\mathcal{X}^{\prime})$ are
chosen such that we have an equivalence
$\mathcal{X}^{\prime}_{\eta^{\prime}}\cong\mathcal{X}_{\eta}\times_{\eta}\eta^{\prime}$
which induces an equivalence $Z^{\prime}_{\eta^{\prime}}\cong
Z_{\eta}\times_{\eta}\eta^{\prime}$.

We start by repeating the following steps of Proposition 6.1.3 of [Tsc23]:

  * •

We choose the appropriate base change
$S^{\prime}\coloneqq\operatorname{Spec}R^{\prime}\to S$ for some discrete
valuation ring $R^{\prime}$, which corresponds to placing the triangle
$\operatorname{trop}(X_{0})$ at a suitable height within the cone
$\operatorname{trop}(X)$ so that the intersection vertices
$\operatorname{trop}(Z)\cap\operatorname{trop}(X_{0})$ are integral (see proof
of Proposition 6.1.3 of [Tsc23] for more details).

  * •

inductively construct an element

$(t_{1},\dots,t_{n+1})=(f_{1}u^{g_{1}},\dots,f_{n+1}u^{g_{n+1}})\in\mathbb{A}^{n+1},$
(4.4.1)

where $u$ is the uniformising parameter of $R^{\prime}$ and
$f_{i}\in(R^{\prime})^{\times}$, i.e. they are units in $R^{\prime}$. This,
paired with a choice of unbroken pair $(A,B)$ for $n+1$, will determine a
morphism $\xi\colon S^{\prime}\to C[A,B]$ such that the pullback
$\xi^{*}X[A,B]$ defines the required family $\mathcal{X}^{\prime}$.

The equation (4.4.1) does not on its own determine an element of
$\mathfrak{X}^{\prime}$ (unlike $\mathfrak{X}$). In order to construct a
stable extension for $(Z,\mathcal{X})$, we still need to make a suitable
choice of sets $A$ and $B$. As Proposition 6.1.3 of [Tsc23] shows, the maximal
choices $A=\\{1,\dots,n\\}$ and $B=\\{2,\dots,n+1\\}$ would work, but these
are not the only possible choices, as in this construction we are not forced
to blow up both $x$ and $y$ along every basis directions. We will see that for
this reason the choice of stable extension is not necessarily unique.

Given different families of subschemes $Z$, we will describe the possible
choices of $A$ and $B$ which give rise to stable extensions
$(Z^{\prime}_{\eta_{0}^{\prime}},\mathcal{X}^{\prime}_{\eta_{0}^{\prime}})$.
Recall, from the proof of Proposition 6.1.3 in [Tsc23], that the subscheme
$Z^{\prime}$ is a union of irreducible components $Z^{\prime}_{i}$ and that
$u$ was chosen such that each $Z^{\prime}_{i}$ can be written locally in terms
of its $x,y$ and $z$ coordinates as

$\\{(c_{i,1}u^{e_{i,1}},c_{i,2}u^{e_{i,2}},c_{i,3}u^{e_{i,3}})\\},$ (4.4.2)

for some $e_{i,j}\in\mathbb{Z}$ and $c_{i,j}\in(R^{\prime})^{\times}$. As
$Z^{\prime}$ is a flat family given by the above expression, it must satisfy
the equations

$\displaystyle x$ $\displaystyle=c_{i,1}u^{e_{i,1}},$ $\displaystyle y$
$\displaystyle=c_{i,2}u^{e_{i,2}},$ (4.4.3) $\displaystyle z$
$\displaystyle=c_{i,3}u^{e_{i,3}}$

also over the closed point. Part of our stability conditions is to require
subschemes to be smoothly supported, so, if more than one element of the set
$\\{e_{i,1},e_{i,2},e_{i,3}\\}$ is nonzero, then
$Z^{\prime}_{\eta^{\prime}_{0}}$ must be supported in a component blown up
along the vanishing of both sides of the relevant equations (4.4.3). This
imposes some restrictions on the $A$ and $B$ we can choose given a certain
$Z$, but as we will see there may still be choices to make.

First, we discuss limits where the choice is unique. If $e_{i,1}$ and
$e_{i,3}$ are nonzero but $e_{i,2}=0$, then the support of
$(Z^{\prime}_{i})_{\eta^{\prime}_{0}}$ lies in the $\pi^{*}(Y_{1}\cap
Y_{3})^{\circ}$ locus. In our construction, there is only one way of expanding
a $\Delta$-component in this locus which will contain the support of
$(Z^{\prime}_{i})_{\eta^{\prime}_{0}}$ in its interior, namely that which in
the localisation is given by blowing up the ideal $\langle
x,u^{e_{i,1}}\rangle$. There is exactly one $j$ such that $t_{1}\cdots
t_{j}=cu^{e_{i,1}}$, for some $c\in(R^{\prime})^{\times}$, where $t_{k}$ are
as in the expression (4.4.1). It follows that $j\in A$.

Similarly, if $e_{i,2}$ and $e_{i,3}$ are nonzero but $e_{i,1}=0$, then the
support of $(Z^{\prime}_{i})_{\eta^{\prime}_{0}}$ lies in the
$\pi^{*}(Y_{2}\cap Y_{3})^{\circ}$ locus. Again, there is only one way of
expanding a $\Delta$-component in this locus such that the flat limit of
$(Z^{\prime}_{i})_{\eta^{\prime}}$ is contained in the interior of this
$\Delta$-component. In the localisation, this is given by the blow-up of the
ideal $\langle y,u^{e_{i,2}}\rangle$. Now, as in the previous example, there
is exactly one $j$ such that $t_{j}\cdots t_{n+1}=cu^{e_{i,2}}$, for some
$c\in(R^{\prime})^{\times}$, where $t_{k}$ are as in the expression (4.4.1)
(the values of $c$ and $j$ may, of course, differ from those in the first
example). It follows that $j\in B$.

Finally, if $e_{i,1}$, $e_{i,2}$ and $e_{i,3}$ are all nonzero then the
support of $(Z^{\prime}_{i})_{\eta^{\prime}_{0}}$ lies in the
$\pi^{*}(Y_{1}\cap Y_{2}\cap Y_{3})$ locus which is pulled back from a
codimension 2 locus in $X_{0}$. This implies that in order for
$(Z^{\prime}_{i})_{\eta^{\prime}_{0}}$ to be smoothly supported, it must be
contained in a $\mathbb{P}^{1}\times\mathbb{P}^{1}$ bubble, given by blowing
up both ideals $\langle x,u^{e_{i,1}}\rangle$ and $\langle
y,u^{e_{i,2}}\rangle$. If $j$ is such that $t_{1}\cdots t_{j}=cu^{e_{i,1}}$
and $k$ is such that $t_{k}\cdots t_{n+1}=cu^{e_{i,2}}$, then we must include
both $j\in A$ and $k\in B$.

Now, on the other hand, if $e_{i,1}$ and $e_{i,2}$ are nonzero but
$e_{i,3}=0$, then the support of $(Z^{\prime}_{i})_{\eta^{\prime}_{0}}$ lies
in the $\pi^{*}(Y_{1}\cap Y_{2})^{\circ}$ locus and there are several possible
ways to expand out a component in this locus which would contain
$(Z^{\prime}_{i})_{\eta^{\prime}_{0}}$ in its interior. Indeed,
$(Z^{\prime}_{i})_{\eta^{\prime}_{0}}$ may lie in a $\Delta_{1}$\- or
$\Delta_{2}$-component of pure type or in a component of mixed type
$\Delta_{1}=\Delta_{2}$. In other words, if $j$ is such that $t_{1}\cdots
t_{j}=cu^{e_{i,1}}$, for some $c\in(R^{\prime})^{\times}$ and therefore
$t_{j+1}\cdots t_{n+1}=du^{e_{i,2}}$, for some $d\in(R^{\prime})^{\times}$,
then we may pick any of the following three options

$\displaystyle j\in A\quad\textrm{and}\quad j+1\notin B,$ $\displaystyle
j\notin A\quad\textrm{and}\quad j+1\in B,$ $\displaystyle j\in
A\quad\textrm{and}\quad j+1\in B.$

Of course, whether or not $j$ and $j+1$ are contained in $A$ and $B$ will
depend also on the other $Z_{i}$ and is constrained by the need for $(A,B,n)$
to be unbroken, so for certain $Z$, some of these choices are removed.

By making compatible choices as above, we may choose an extension
$(Z^{\prime}_{\eta_{0}},\mathcal{X}^{\prime}_{\eta_{0}})$ such that at least
one point of the support of $Z^{\prime}_{\eta_{0}}$ is in the union
$(\Delta_{1}^{(i)})^{\circ}\cup(\Delta_{2}^{(n+1-i)})^{\circ}$ for every $i$
for which either of these components exists in
$\mathcal{X}^{\prime}_{\eta_{0}}$. By Theorem 4.2, this limit is SWS stable.

This shows that if $(Z_{\eta},\mathcal{X}_{\eta})$ is pulled back from a fibre
above a point $(t_{1},\dots,t_{n+1})$ in some $C[A,B]$ whose entries are all
invertible, then $(Z_{\eta},\mathcal{X}_{\eta})$ has an SWS stable extension.
See Corollary 5.13 for a proof that there exists an extension if
$\mathcal{X}_{\eta}$ is a modified special fibre. ∎

######  Corollary 4.13.

The stack $\mathfrak{N}^{m}_{\operatorname{LW}}$ is universally closed.

###### Proof.

As SWS stable pairs must be LW stable, this follows immediately. ∎

######  Remark 4.14.

The proof of Proposition 4.12 shows how the stacks
$\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$ fail to be separated by highlighting
several possible choices of limit representatives. We should note also that it
is possible for such a pair
$(Z^{\prime}_{\eta_{0}},\mathcal{X}^{\prime}_{\eta_{0}})$ to have finite
automorphisms while having expanded $\Delta$-components in
$\mathcal{X}^{\prime}_{\eta_{0}}$ which contain no point of the support of
$Z^{\prime}_{\eta_{0}}$. This may happen for example if, in the expression
(4.4.2), both $e_{i,1}$ and $e_{i,3}$ are nonzero but $e_{i,2}=0$. Then, in
order for $Z^{\prime}_{\eta_{0}}$ to be smoothly supported in
$\mathcal{X}^{\prime}_{\eta_{0}}$, we must let $j\in A$ (using the notation of
the proof) so as to expand out the component $\Delta_{1}^{(j)}$; but we may
also let $j+1\in B$ so as to have the component $\Delta_{2}^{(n+1-j)}$
expanded out in $\mathcal{X}^{\prime}_{\eta_{0}}$, even if the latter contains
no point of the support. As the same $\mathbb{G}_{m}$ acts on
$\Delta_{1}^{(j)}$ and $\Delta_{2}^{(n+1-j)}$, this will not add any
automorphisms. In fact, this is exactly the type of blow-up we made in
[Tsc23].

##  5 Constructing separated stacks.

We are now in a position to construct separated stacks. As mentioned earlier,
there are two ways of doing this. Firstly, one can avoid making any choices
and instead identify all possible limits of a given family in the stacks
$\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$. We denote these by
$\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}$ and
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}$ respectively. This procedure
is in a sense more general, as it avoids making choices. As we will see, this
comes at the cost of these stacks no longer being algebraic. The second way to
obtain separated stacks is to make a systematic choice of representative for a
given limit. This will be done by adding an additional stability condition.

###  5.1 Non-algebraic proper stacks

#### Non-unique associated pairs.

We recall the notion of _associated pairs_ from [Tsc23].

######  Definition 5.1.

Let $(Z,\mathcal{X})\in\mathfrak{N}^{m}_{\operatorname{SWS}}(k)$ (or
$\mathfrak{N}^{m}_{\operatorname{LW}}(k)$), where $k$ is an algebraically
closed field as before. We will say that a pair $(Z,\mathcal{X})$ is an
associated pair for a configuration of vertices in
$\operatorname{trop}(X_{0})$ (or rays through these vertices in
$\operatorname{trop}(X)$) if there is a 1-to-1 correspondence between these
vertices and the non-empty bubbles in $(Z,\mathcal{X})$ given as follows: to
each such vertex $(e_{i,1},e_{i,2},e_{i,3})\in\operatorname{trop}(X_{0})$
corresponds a non-empty bubble blown-up along vanishing of both sides of the
equations (4.4.3).

The problem of non-separatedness comes from there being more than one
associated pair in $\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$ for a given configuration of vertices
in $\operatorname{trop}(X_{0})$. For example, in Figure 4, we see that there
are three possible stable fibre-subscheme pairs associated to the tropical
picture on the left. In other words, there are several possible blow-ups of
$X_{0}$ which create a bubble corresponding to the vertex
$\operatorname{trop}(Z)$ in the picture on the left. Moreover, there are
exactly three such blow-ups for which this bubble being the only component
containing a point of the support of $Z$ creates a stable pair. The red point
in each of the three geometric pictures on the right corresponds to the
support of the subscheme $Z$.

$\operatorname{trop}(Z)$$Y_{1}$$Y_{3}$$Y_{2}$

$Y_{1}$$Y_{2}$$Y_{3}$

$Y_{1}$$Y_{2}$$Y_{3}$

$Y_{1}$$Y_{2}$$Y_{3}$

Figure 4: Three different stable associated pairs for the same tropical
picture.

#### The stacks $\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}$ and
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}$.

Let $S\coloneqq\operatorname{Spec}R\to C$ for some valuation ring $R$ as
before and let $\eta$ and $\eta_{0}$ denote the generic and closed points of
$S$. We construct the separated stacks
$\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}$ and
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}$ from the stacks
$\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$ by introducing additional equivalences
on these stacks. It is sufficient to define these equivalences over the scheme
$S$ as we already have a well-defined notion of equivalence up to base change
on these stacks.

Let
$(Z_{\eta},\mathcal{X}_{\eta})\in\mathfrak{N}^{m}_{\operatorname{SWS}}(\eta)$
(or $\mathfrak{N}^{m}_{\operatorname{LW}}(\eta)$), where all basis directions
are invertible at the point $\eta$. Then $(Z_{\eta},\mathcal{X}_{\eta})$ will
also be an element of $\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}(\eta)$
(or $\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}(\eta)$). We then define
to be equivalent in $\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}$ (or
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}$) all stable (for the
respective stability conditions) extensions of this pair. In other words, any
two pairs in $\mathfrak{N}^{m}_{\operatorname{SWS}}(k)$ (or
$\mathfrak{N}^{m}_{\operatorname{LW}}(k)$) which are associated pairs for the
same configuration of vertices in $\operatorname{trop}(X_{0})$ (or rays in
$\operatorname{trop}(X)$) must belong to the same equivalence class in
$\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}(k)$ (or
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}(k)$).

The only choices here come from the construction of the stack
$\mathfrak{X}^{\prime}$, but all subsequent choices are avoided. This,
however, implies that the objects of the stack are equivalence classes of
pairs $(Z,\mathcal{X})$, where for two representatives $(Z,\mathcal{X})$ and
$(Z^{\prime},\mathcal{X}^{\prime})$ of a given equivalence class, there may
exist no isomorphism of schemes $\mathcal{X}\cong\mathcal{X}^{\prime}$.
Defining this more canonical stack therefore comes at the cost of our stack
being no longer algebraic. This parallels the canonical choice of underlying
stack in Kennedy-Hunt’s construction of a logarithmic Quot scheme [Ken23].

######  Theorem 5.2.

The stacks $\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}$ and
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}$ have finite automorphisms
and are proper.

###### Proof.

They have finite automorphisms because the stacks
$\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$ have finite automorphisms by the same
argument as Lemma 6.1.10 of [Tsc23]. Passing to the equivalence classes of
associated pairs does not add automorphisms, since the equivalences being
added are not isomorphisms. The universal closure follows directly from the
universal closure of $\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$ in Proposition 4.12 and Corollary 4.13,
and from Corollary 5.13. It remains to prove that the stacks are separated.

Let $S\coloneqq\operatorname{Spec}R\to C$, where $R$ is a discrete valuation
ring with uniformising parameter $u$. Let $\eta$ denote the generic point of
$S$ and $\eta_{0}$ its closed point. Now, assume that there are two pairs
$[(Z,\mathcal{X})],[(Z^{\prime},\mathcal{X}^{\prime})]\in\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}(S)$
(or $\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}(S)$) such that
$[(Z_{\eta},\mathcal{X}_{\eta})]\cong[(Z^{\prime}_{\eta},\mathcal{X}^{\prime}_{\eta})]$.
We will show that it must follow that
$[(Z_{\eta_{0}},\mathcal{X}_{\eta_{0}})]\cong[(Z^{\prime}_{\eta_{0}},\mathcal{X}^{\prime}_{\eta_{0}})]$.

We consider the representatives $(Z,\mathcal{X})$ and
$(Z^{\prime},\mathcal{X}^{\prime})$ of the above equivalence classes. As in
the the proof of Proposition 4.12, we may assume that $S$ is chosen so that
the $i$-th irreducible component of $Z$ is given in terms of its local
coordinates $x,y$ and $z$ by

$\\{(c_{i,1}u^{e_{i,1}},c_{i,2}u^{e_{i,2}},c_{i,3}u^{e_{i,3}})\\},$ (5.1.1)

and the $i$-th irreducible component of $Z^{\prime}$ is given in terms of its
local coordinates $x,y$ and $z$ by

$\\{(d_{i,1}u^{f_{i,1}},d_{i,2}u^{f_{i,2}},d_{i,3}u^{f_{i,3}})\\}.$ (5.1.2)

Since the equivalences of the stack act trivially on $x,y$ and $z$ and we know
that
$(Z_{\eta},\mathcal{X}_{\eta})\cong(Z^{\prime}_{\eta},\mathcal{X}^{\prime}_{\eta})$,
it must therefore follow that $Z$ and $Z^{\prime}$ have the same number of
irreducible components. Moreover, if these components are labelled in a
compatible way, then $c_{i,1}=d_{i,1}$ and $e_{i,1}=f_{i,1}$ for all $i$. This
is independent of the choices of representative $Z$ and $Z^{\prime}$, as the
valuation vectors $(e_{i,1},e_{i,2},e_{i,3})$ and $(f_{i,1},f_{i,2},f_{i,3})$
are precisely the data of the tropicalisation of $Z$ and $Z^{\prime}$ and this
is constant across the equivalence classes.

Now, as before, the fact that $Z$ and $Z^{\prime}$ must be $S$-flat families
in $\mathcal{X}$ and $\mathcal{X}^{\prime}$ respectively means that each
$Z_{i}$ and $Z_{i}^{\prime}$ component must satisfy the equations

$\displaystyle x$ $\displaystyle=c_{i,1}u^{e_{i,1}},$ $\displaystyle y$
$\displaystyle=c_{i,2}u^{e_{i,2}},$ $\displaystyle z$
$\displaystyle=c_{i,3}u^{e_{i,3}},$

also above the closed point. If more than one element of the set
$\\{e_{i,1},e_{i,2},e_{i,3}\\}$ is nonzero, then $Z_{i}$ and $Z_{i}^{\prime}$
must be supported in a component blown up along $\langle
x,cu^{e_{i,1}}\rangle$ and $\langle y,c^{\prime}u^{e_{i,2}}\rangle$ over the
closed point $\eta_{0}$, for some $c,c^{\prime}\in R^{\times}$. But this tells
us exactly that the nonempty bubbles of the pairs
$(Z_{\eta_{0}},\mathcal{X}_{\eta_{0}})$ and
$(Z^{\prime}_{\eta_{0}},\mathcal{X}^{\prime}_{\eta_{0}})$ must correspond to
the same points in the tropicalisation and therefore these two pairs must
belong to the same equivalence class. ∎

######  Remark 5.3.

As mentioned above, the stacks
$\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}$ and
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}$ mirror Kennedy-Hunt’s
underlying stack construction of a logarithmic Quot scheme. We should note,
however, that restricting the construction of [Ken23] to the case of Hilbert
schemes of points, though similar, would not yield either of the stacks
$\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}$ and
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}$. This is because some
choices were made in the way we constructed $X[A,B]$, and
$\mathfrak{X}^{\prime}$ does not contain every possible expansion of $X$.
Kennedy-Hunt’s construction precludes any such choices. Let us denote for now
by $\mathfrak{Y}\to C$ the underlying stack of the restriction of the
logarithmic Quot scheme construction to the case of Hilbert schemes of points.
For any algebraically closed field $k$, an object of $\mathfrak{Y}(k)$ is an
equivalence class. While there is a bijection between the sets of equivalence
classes $|\mathfrak{Y}(k)|$ and
$|\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}(k)|$ or
$|\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}(k)|$ of the respective
stacks, some representatives of the equivalence class defining an object of
$\mathfrak{Y}(k)$ do not exist in the equivalence class defining the
corresponding object in $\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}(k)$
or $\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}(k)$.

Finally, we would like to comment on the fact that we do not assert an
isomorphism between the stacks
$\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}$ and
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}$ here. This is because the
results we used to establish equivalence of stacks require the stacks to be
algebraic. We do, in fact, expect these stacks to be isomorphic but some
further work is needed to show this.

###  5.2 Choices of proper algebraic stacks.

As mentioned before, the second way we have of making separated stacks is to
make a systematic choice among all pairs
$(Z,\mathcal{X})\in\mathfrak{N}^{m}_{\operatorname{SWS}}(k)$ (or
$\mathfrak{N}^{m}_{\operatorname{LW}}(k)$) associated to the same
configuration of vertices in $\operatorname{trop}(X_{0})$. For example, in
Figure 4, we would need to add a stability condition which allows exactly one
of the pairs represented by the three geometric pictures on the right to be
stable. These choices effectively cut out proper substacks of
$\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$.

#### Additional stability condition for uniqueness of limits.

In order to make these choices of limit representatives, we may declare that
certain fibres $\mathcal{X}\in\mathfrak{X}^{\prime}$ contain no stable
subschemes and we may also decide that certain bubbles in given fibres contain
no points of the support. The Maulik-Ranganathan stability condition is close
to this description. The starting point for their construction is to look at
the image of a given subscheme $Z$ under the tropicalisation map, given by a
collection of vertices in the tropicalisation of $X_{0}$, and then construct
an expansion around the bubbles introduced by these new vertices,
corresponding to our notion of associated pairs. Implicitly, choices are made
at this stage to exclude certain superfluous fibres. We must therefore emulate
this with our stability condition. Moreover, in the modified special fibres
they do construct, it is necessary to allow certain configurations of points
and not others, hence labelling certain bubbles as tube components and
introducing the concept of Donaldson-Thomas stability with respect to these
labellings. We describe suitable additions to the stability conditions
previously discussed and extend Maulik-Ranganathan stability to our situation
with the following definitions.

######  Definition 5.4.

Let $\mathcal{X}\in\mathfrak{X}^{\prime}(k)$ and let $Z$ be a length $m$ zero-
dimensional subscheme in $\mathcal{X}$. A _tube labelling_ of $\mathcal{X}$ is
a choice of a collection of bubbles in $\mathcal{X}$ which we call _tube
components_. Given such a labelling, the pair $(Z,\mathcal{X})$ is Donaldson-
Thomas stable (abbreviated DT stable) if it satisfies the following condition:
a bubble in $\mathcal{X}$ contains no point of the support of $Z$ if and only
if it is a tube component.

######  Definition 5.5.

Let $\alpha$ denote a collection of equivalence classes of fibres in
$\mathfrak{X}^{\prime}(k)$ and let $\beta$ denote a collection of tube
labellings on the equivalence classes of fibres remaining after the fibres
$\alpha$ have been removed. Let $\mathcal{X}\in\mathfrak{X}^{\prime}(k)$. We
say that the pair $(Z,\mathcal{X})$, where $Z$ is a length $m$ zero-
dimensional subscheme in $\mathcal{X}$, is $(\alpha,\beta)$-stable if
$\mathcal{X}$ is not one of the fibres in $\alpha$ and the pair
$(Z,\mathcal{X})$ is DT stable.

We will only want to consider $(\alpha,\beta)$-stability as a restriction of
SWS or LW stability. We denote by
$\mathfrak{N}^{m}_{\operatorname{SWS},(\alpha,\beta)}$ and
$\mathfrak{N}^{m}_{\operatorname{LW},(\alpha,\beta)}$ the stacks
$\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$ restricted to their
$(\alpha,\beta)$-stable loci. As we will see with the next two results, an
appropriate choice of $(\alpha,\beta)$-stability condition will give us unique
limits and allow us to build proper stacks. First, let us define some useful
terminology.

######  Definition 5.6.

We say that a pair $(\alpha,\beta)$ is an almost proper SWS stability
condition if it is chosen so that for each configuration of points in
$\operatorname{trop}(X_{0})$ there exists a unique associated pair
$(Z,\mathcal{X})\in\mathfrak{N}^{m}_{\operatorname{SWS},(\alpha,\beta)}(k)$,
where $\mathcal{X}$ is pulled back from a modified special fibre. The
corresponding definition of almost proper LW stability condition can be made
in a similar way.

######  Proposition 5.7.

Let $S\coloneqq\operatorname{Spec}R$ for some discrete valuation ring $R$ and
let $\xi\colon S\to C[A,B]$ be a morphism that maps the generic point $\eta$
of $S$ to a vector in $C[A,B]$ whose entries are all invertible, for some
unbroken $(A,B,n)$. Now, assume that $(\alpha,\beta)$ is an almost proper SWS
or LW stability condition and let $\mathcal{X}\coloneqq\xi^{*}X[A,B]$. If
$(Z_{\eta}\mathcal{X}_{\eta})\in\mathfrak{N}^{m}_{\operatorname{SWS},(\alpha,\beta)}(\eta)$
(or $\mathfrak{N}^{m}_{\operatorname{LW},(\alpha,\beta)}(\eta)$), then
$(Z_{\eta},\mathcal{X}_{\eta})$ has a unique extension in
$\mathfrak{N}^{m}_{\operatorname{SWS},(\alpha,\beta)}$ (or
$\mathfrak{N}^{m}_{\operatorname{LW},(\alpha,\beta)}$).

###### Proof.

This follows directly from the proof of Proposition 4.12. Indeed, given such a
pair $(Z_{\eta},\mathcal{X}_{\eta})$, we recall that, up to some finite base
change, each irreducible component of $Z_{\eta}$ can be written locally in
terms of its $x,y$ and $z$ coordinates as

$\\{(c_{i,1}u^{e_{i,1}},c_{i,2}u^{e_{i,2}},c_{i,3}u^{e_{i,3}})\\},$

for some $e_{i,j}\in\mathbb{Z}$ and $c_{i,j}\in R^{\times}$. Moreover, we
showed that if more than one element of the set
$\\{e_{i,1},e_{i,2},e_{i,3}\\}$ is nonzero, the closure of $Z_{\eta}$ in any
stable extension of $(Z_{\eta},\mathcal{X}_{\eta})$ must be supported in a
component blown up along the vanishing of both sides of the relevant equations
(4.4.3). Now, we note that each of these triples $(e_{i,1},e_{i,2},e_{i,3})$
corresponds to a point in $\operatorname{trop}(X_{0})$. By the proof of
Proposition 4.12, any associated pair for $\operatorname{trop}(Z_{\eta})$
gives an extension of $(Z_{\eta},\mathcal{X}_{\eta})$. Since $(\alpha,\beta)$
is an almost proper SWS stability condition, we know that there exists a
unique $(\alpha,\beta)$-stable associated pair providing such an extension. ∎

We have therefore proven that, if $(\alpha,\beta)$ is an almost proper SWS (or
LW) stability condition, families whose generic fibre is pulled back from a
fibre in $X[A,B]$ over a point $(t_{1},\dots,t_{n+1})$ whose entries are all
nonzero have a unique $(\alpha,\beta)$-stable limit.

#### Compatibility condition for properness.

In order to ensure that the stacks
$\mathfrak{N}^{m}_{\operatorname{SWS},(\alpha,\beta)}$ and
$\mathfrak{N}^{m}_{\operatorname{LW},(\alpha,\beta)}$ are proper, we need one
more compatibility condition to hold. The following example illustrates how
the stacks may fail to be proper if we do not add this extra condition.

_Example of incompatible limits._ Let $(\alpha,\beta)$ be an almost proper LW
stability condition. Let the equivalence class in
$\mathfrak{N}^{m}_{\operatorname{LW},(\alpha,\beta)}$ associated to
$\operatorname{trop}(X_{0})$ with one vertex added to the $Y_{1}\cap Y_{2}$
edge be given by expanding one $\Delta_{1}$-component of pure type. Now, we
decide that the equivalence class associated to $\operatorname{trop}(X_{0})$
with two vertices added to the $Y_{1}\cap Y_{2}$ edge will be given by
expanding two $\Delta_{2}$-components of pure type. These determine the unique
associated pairs for these tropical configurations. See Figure 5. Now let us
take a length 2 zero-dimensional subscheme in the first fibre whose support
consists of a point in $\Delta_{1}^{\circ}\cap\pi^{*}(Y_{1}\cap
Y_{2})^{\circ}$ and a point in $Y_{2}^{\circ}$. This is LW and
$(\alpha,\beta)$-stable for our choice of $(\alpha,\beta)$. We may now study
the limit of this subscheme as the point in $Y_{2}^{\circ}$ tends towards
$\pi^{*}(Y_{1}\cap Y_{2})$, i.e. as its $x$ coordinate tends to zero. If the
stack $\mathfrak{N}^{m}_{\operatorname{LW},(\alpha,\beta)}$ were proper it
would contain this limit, but it does not. Indeed, our second fibre has the
same associated tropical configuration as this limit and by the almost proper
condition it must be the unique fibre in
$\mathfrak{N}^{m}_{\operatorname{LW},(\alpha,\beta)}$ up to equivalence which
does, but it is clearly not a limit for this pair. This breaks universal
closure.

Figure 5: The picture on the right is not a limit of the one on the left.

We recall the following definitions from [Tsc23].

######  Definition 5.8.

Let S be a $C$-scheme and $\mathcal{X}\in\mathfrak{X}(S)$. The equivalence
class $[\operatorname{trop}(\mathcal{X})]$ is defined as follows. For any
$C$-scheme $S^{\prime}$ and any
$\mathcal{X}^{\prime}\in\mathfrak{X}(S^{\prime})$, we say
$\operatorname{trop}(\mathcal{X}^{\prime})$ belongs to the equivalence class
$[\operatorname{trop}(\mathcal{X})]$ if and only if $\mathcal{X}$ and
$\mathcal{X}^{\prime}$ are equivalent in $\mathfrak{X}$.

By convention, if $\operatorname{trop}(\mathcal{X})$ is not well-defined (for
example, if we have only specified a family in $X[A,B]$ from which it is
pulled back but not a morphism $S\to C[A,B]$), we may take
$\operatorname{trop}(\mathcal{X})$ to be any element of the equivalence class
$[\operatorname{trop}(\mathcal{X})]$.

For an unbroken pair $(A,B,n)$ and a set $I\subseteq[n+1]$, let $X[A,B]_{I}$
denote the restriction of $X[A,B]$ to the locus where $t_{i}=0$ for all $i\in
I$.

######  Definition 5.9.

Let
$(Z_{\eta},\mathcal{X}_{\eta})\in\mathfrak{N}^{m}_{\operatorname{SWS},(\alpha,\beta)}(\eta)$
(or $\mathfrak{N}^{m}_{\operatorname{LW},(\alpha,\beta)}(\eta)$) be any pair
over the generic point of some $S\coloneqq\operatorname{Spec}R$, for some
discrete valuation ring $R$ as before. Moreover, let $\mathcal{X}_{\eta}$ be a
restriction of $\mathcal{X}\coloneqq\xi^{*}X[A,B]_{I}$ for some nonempty set
$I$, i.e. $\mathcal{X}_{\eta}$ is pulled back from some modified special
fibre. If, for any associated pair
$(Z^{\prime}_{\eta^{\prime}_{0}},\mathcal{X}^{\prime}_{\eta^{\prime}_{0}})$ of
$\operatorname{trop}(Z_{\eta})$ in $\mathfrak{N}^{m}_{\operatorname{SWS}}$ (or
$\mathfrak{N}^{m}_{\operatorname{LW}}$), the tropicalisation
$\operatorname{trop}(\mathcal{X}^{\prime}_{\eta^{\prime}_{0}})$ is a
subdivision of a representative of the equivalence class
$[\operatorname{trop}(\mathcal{X}_{\eta})]$ (as in Definition 5.8), then we
say that $\mathfrak{N}^{m}_{\operatorname{SWS},(\alpha,\beta)}$ (or
$\mathfrak{N}^{m}_{\operatorname{LW},(\alpha,\beta)}$) is _tropically
compatible_.

Now, we may define the appropriate compatibility condition for
$(\alpha,\beta)$-stability.

######  Definition 5.10.

Let $(\alpha,\beta)$ define an almost proper SWS stability condition. If the
corresponding stack $\mathfrak{N}^{m}_{\operatorname{SWS},(\alpha,\beta)}$ is
tropically compatible, we say that $(\alpha,\beta)$ defines a proper SWS
stability condition (abbreviated PSWS stability). A similar definition can be
made for LW stability. If $(\alpha,\beta)$ is a proper LW stability condition
and the pair $(Z,\mathcal{X})$ is LW and $(\alpha,\beta)$-stable, then we say
it is Maulik-Ranganathan stable (abbreviated MR stable) for the given choice
$(\alpha,\beta)$.

######  Proposition 5.11.

Let $(\alpha,\beta)$ be some choice of stability condition. The corresponding
stack $\mathfrak{N}^{m}_{\operatorname{SWS},(\alpha,\beta)}$ is proper if and
only if $(\alpha,\beta)$ is a PSWS stability condition.

###### Proof.

This follows from Proposition 6.1.8 of [Tsc23], since we saw in the proof of
that proposition that there are two conditions needed to show the moduli stack
is proper:

  * •

existence and uniqueness of the associated pair, which is given two us by the
almost proper SWS stability condition $(\alpha,\beta)$, and

  * •

the tropically compatible property, which is given to us by the fact that
$(\alpha,\beta)$ is moreover a proper SWS stability condition.

∎

######  Corollary 5.12.

The stack $\mathfrak{N}^{m}_{\operatorname{LW},(\alpha,\beta)}$ is proper if
and only if $(\alpha,\beta)$ is a proper LW stability condition.

###### Proof.

This follows immediately from the proof of Proposition 5.11. ∎

_Notation._ In the following we shall denote by
$\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}$ a stack
$\mathfrak{N}^{m}_{\operatorname{SWS},(\alpha,\beta)}$ where $(\alpha,\beta)$
is a proper SWS stability condition, and by
$\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}$ a stack
$\mathfrak{N}^{m}_{\operatorname{LW},(\alpha,\beta)}$ where $(\alpha,\beta)$
is a proper LW stability condition.

The stacks $\mathfrak{M}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{M}^{m}_{\operatorname{LW}}$ from [Tsc23] are examples of stacks
$\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}$ and
$\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}$. Indeed, the choice
$(\alpha,\beta)$ which defines these stacks clearly satisfies the conditions
of a proper SWS (or LW) stability condition.

######  Corollary 5.13.

Let $S$ and $\eta$ as before and let $(Z_{\eta},\mathcal{X}_{\eta})$ be an
element of $\mathfrak{N}^{m}_{\operatorname{SWS}}(\eta)$,
$\mathfrak{N}^{m}_{\operatorname{LW}}(\eta)$,
$\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}(\eta)$ or
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}(\eta)$ such that
$\mathcal{X}_{\eta}$ is a modified special fibre. Then
$(Z_{\eta},\mathcal{X}_{\eta})$ has a stable extension with respect to the
relevant stability condition in the relevant stack.

###### Proof.

In the non-separated stacks $\mathfrak{N}^{m}_{\operatorname{SWS}}$ and
$\mathfrak{N}^{m}_{\operatorname{LW}}$, we allow for all limits. In
particular, each stack will contain all the associated pairs for
$\operatorname{trop}(Z_{\eta})$ which are compatible with $\mathcal{X}_{\eta}$
and stable for their respective stability conditions.

In the case of $\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}(\eta)$ and
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}(\eta)$, we do not make any
choices, so for any such $(Z_{\eta},\mathcal{X}_{\eta})$ all possible
associated pairs of $\operatorname{trop}(Z_{\eta})$ form an equivalence class
in $\overline{\mathfrak{N}^{m}_{\operatorname{SWS}}}(\eta)$ or
$\overline{\mathfrak{N}^{m}_{\operatorname{LW}}}(\eta)$. In particular, there
will be a representative of this equivalence class which is compatible with
$\mathcal{X}_{\eta}$ in the sense of Definition 5.10. ∎

######  Theorem 5.14.

The stacks $\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}$ and
$\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}$ are Deligne-Mumford and
proper over $C$.

###### Proof.

Properness follows from Proposition 5.11 and Corollary 5.12. They are Deligne-
Mumford because SWS and LW stability ensure finite automorphisms. ∎

#### Remark about compatibility.

Here, we point out a subtlety about the compatibility conditions discussed
above. Let $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ be two fibres of
$\mathfrak{X}^{\prime}$ over a closed point. Assume that $\mathcal{X}_{1}$ is
pulled back from some $X[A,B]_{J_{1}}$ and $\mathcal{X}_{2}$ is pulled back
from some $X[A^{\prime},B^{\prime}]_{J_{2}}$, where $|J_{1}|\leq|J_{2}|$. We
denote by $W_{1}\subseteq\mathcal{X}_{1}$ and $W_{2}\subseteq\mathcal{X}_{2}$
the union of all irreducible components in each fibre which are not tubes
(this includes the $Y_{i}$ components). We say that $(\alpha,\beta)$ defines a
strong compatibility condition if the following holds. If
$\pi_{*}(W_{2})\subset\overline{\pi_{*}(W_{1})}\subset X_{0}$, then there
exists a fibre $\mathcal{X}_{3}$ equivalent to $\mathcal{X}_{2}$ in
$\mathfrak{X}^{\prime}$ such that $\mathcal{X}_{3}$ is pulled back from
$X[A,B]_{J_{3}}$ for some $J_{3}$ with $J_{1}\subseteq J_{3}$. This condition
certainly implies tropical compatibility, and naively may seem like the right
restriction to impose, but it is actually stronger than necessary. Indeed, the
tropically compatible condition does not necessarily imply strong
compatibility. As shown in Remark 5.15, strong compatibility is not necessary
to construct a proper stack, as in certain cases, subschemes cannot tend
towards a certain limit.

######  Remark 5.15.

Let $(Z_{\eta},\mathcal{X}_{\eta})$ be a pair consisting of a length $m$ zero-
dimensional subscheme $Z_{\eta}$ in a modified special fibre
$\mathcal{X}_{\eta}$. Assume that the pair is either SWS or LW stable. Let $P$
be a point of the support which lies in some irreducible component
$W\subset\mathcal{X}_{\eta}$ and let $V$ be some other irreducible component
in $\mathcal{X}_{\eta}$ which intersects $W$ nontrivially. If there exists no
representative in the equivalence class of $(Z_{\eta},\mathcal{X}_{\eta})$
such that there is a smoothing from the interior of $W$ in
$\mathcal{X}_{\eta}$ to the interior of some $\Delta$-components expanded out
in the $W\cap V$ locus, i.e. such that these $\Delta$-components are equal to
$W$ in $\mathcal{X}_{\eta}$, then there exists no flat family of length $m$
zero-dimensional subschemes such that $(Z_{\eta},\mathcal{X}_{\eta})$ is the
generic fibre and $P$ tends towards $W\cap V$ over the closed point. This can
be seen by studying the equations of the blow-ups.

For example, let $\mathcal{X}_{\eta}$ be a fibre with one expanded
$\Delta_{1}$-component of pure type and no other expanded components. Let
$Z_{\eta}$ be a length 2 zero-dimensional subscheme with one point of its
support, $P_{0}$, lying in the interior of this $\Delta_{1}$-component in the
$\pi^{*}(Y_{1}\cap Y_{2})^{\circ}$ locus and the other point of the support,
$P_{1}$, lying in $Y_{2}^{\circ}$, as in the picture on the left of Figure 5.
In this fibre, there cannot be a $\Delta_{2}$-component which is equal to
$Y_{2}$ and thus there can be no smoothing from the interior of $Y_{2}$ to an
expanded component in the $Y_{2}\cap Y_{3}$ locus. But there also cannot be
any flat family such that $P_{1}$ tends towards $z=0$, as any equation for
$P_{1}$ fixes $z$ and does not set it to be proportional to any variable which
may tend towards zero (like a $y_{1}^{(k)}/y_{0}^{(k)}$ or $t_{i}$ variable).

In Section 5.4, we give a tropical criterion to understand when such
smoothings from one component to another exist.

###  5.3 Relating the proper stacks

We may now show how the stacks
$\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}$ and
$\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}$ relate to each other.

######  Theorem 5.16.

Let $\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}$ and
$\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}$ be two choices given by
the same $(\alpha,\beta)$. This means that we have chosen an MR stability
condition $(\alpha,\beta)$ which also defines a PSWS stability condition (by
Proposition 4.8 this is not the case in general). There then exists an
isomorphism of stacks

$\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}\cong\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}.$

###### Proof.

Similarly to the arguments of Section 6.2 of [Tsc23], it is clear that on the
scheme level we have an inclusion

$H^{m,[A,B]}_{\operatorname{SWS},(\alpha,\beta)}\longhookrightarrow
H^{m,[A,B]}_{\operatorname{LW},(\alpha,\beta)},$

where $H^{m,[A,B]}_{\operatorname{SWS},(\alpha,\beta)}$ and
$H^{m,[A,B]}_{\operatorname{LW},(\alpha,\beta)}$ are the restrictions of
$H^{m}_{[A,B],\operatorname{SWS}}$ and $H^{m}_{[A,B],\operatorname{LW}}$ to
their respective $(\alpha,\beta)$-stable loci. As before, this gives rise to a
morphism of stacks

$f\colon\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}\longrightarrow\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}.$

By our assumptions and Theorem 4.2, for every MR stable pair
$(Z,\mathcal{X})\in\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}(k)$,
where $k$ is an algebraically closed field as before, there exists a
representative of the equivalence class of $(Z,\mathcal{X})$ which is SWS
stable. We therefore have an induced bijection between the sets of equivalence
classes of each stack

$|f|\colon|\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}(k)|\longrightarrow|\mathfrak{N}^{m}_{\operatorname{MR},(\alpha,\beta)}(k)|.$

By Theorem 5.14, $\mathfrak{N}^{m}_{\operatorname{PSWS},(\alpha,\beta)}$ is
separated and Deligne-Mumford and therefore has finite inertia. The morphism
$f$ induces a bijective homomorphism of stabilisers by the same argument as in
the proof of Theorem 6.2.4 of [Tsc23]. By Lemma 6 of [AK16] (Lemma 6.2.3 of
[Tsc23]), this implies that $f$ is representable. Therefore $f$ is an
isomorphism of stacks by Lemma 6.2.1 of [Tsc23]. ∎

As shown in [MR24], all choices of proper Deligne-Mumford stacks obtained for
different $(\alpha,\beta)$, are birational to each other.

###  5.4 Further comments on smoothings

Given a pair $(Z,\mathcal{X})\in\mathfrak{N}^{m}_{\operatorname{SWS}}(k)$ (or
$\mathfrak{N}^{m}_{\operatorname{LW}}(k)$), where $\mathcal{X}$ is pulled back
from a modified special fibre, we would like to understand what deformations
of $(Z,\mathcal{X})$ exist in $\mathfrak{N}^{m}_{\operatorname{SWS}}$ (or
$\mathfrak{N}^{m}_{\operatorname{LW}}$). In particular we would like to study
what smoothings exist from irreducible components of $\mathcal{X}$ to
irreducible components of a second fibre $\mathcal{X}^{\prime}$. This will
allow us to understand explicitly what limits need to be included in the
stacks we construct. We will say there is a _smoothing_ from an irreducible
component $W\subset\mathcal{X}$ to an irreducible component
$W^{\prime}\subset\mathcal{X}^{\prime}$ if there exist a discrete valuation
ring $R$ and a smooth variety $\mathcal{Y}$ over
$S\coloneqq\operatorname{Spec}R$ such that $W$ is equal to the restriction of
$\mathcal{Y}$ to the generic point of $S$ and the restriction of $\mathcal{Y}$
to the closed point of $S$ is a union of irreducible components, where
$W^{\prime}$ is one of these components. The question we seek to answer in
this subsection is the following.

_Given a polyhedral subdivision of $\operatorname{trop}(X_{0})$, what
smoothings exist from the components corresponding to the vertices in
$\operatorname{trop}(X_{0})$_?

The question of smoothings takes place on the level of representatives of
equivalence classes in $\mathfrak{X}^{\prime}$ as it depends on
$\Delta_{i}$-multiplicity (see Definition 3.5). In order to interpret this
information tropically, we enhance $\operatorname{trop}(X_{0})$ with the
notion of $\Delta$-multiplicity.

######  Definition 5.17.

Let $\mathcal{X}\in\mathfrak{X}^{\prime}(k)$ be a modified special fibre and
let $\operatorname{trop}(\mathcal{X})$ be the corresponding polyhedral
subdivision of $\operatorname{trop}(X_{0})$. For each edge of $\Delta$-type 1
or 2 in $\operatorname{trop}(\mathcal{X})$ we may assign to it a positive
integer, called its _tropical $\Delta$-multiplicity_.

######  Remark 5.18.

The tropical $\Delta$-multiplicity of an edge in
$\operatorname{trop}(\mathcal{X})$ is the $\Delta_{i}$-multiplicity common to
all geometric components corresponding to the vertices connected to this edge.
For convenience, here we assign the multiplicity to the edge and not the
vertices, as the edge contains the information of the $\Delta$-type (see
Definition 3.5).

Given a fibre $\mathcal{X}$ as above and any bubble $B_{0}$ in $\mathcal{X}$,
we will show in Proposition 5.19 what smoothings exist from $B_{0}$ to bubbles
$B_{i}$ in blow-ups of $\mathcal{X}$ (possibly after base change). Before
formulating this proposition, we describe the operations on
$\operatorname{trop}(\mathcal{X})$ which can result in smoothings.

#### Allowing for base changes.

As we are working with stacks, we always allow for base changes. Tropically,
this means we may adjust the height of the subdivided triangle
$\operatorname{trop}(\mathcal{X})$ within the cone of the 1-parameter family.
We abuse notation slightly and denote this object after base change by
$\operatorname{trop}(\mathcal{X})$ again. Effectively, this re-scales and
allows us to add more integral points to the triangle. Geometrically, this
corresponds to the phenomenon that we can consider any scheme $X[A,B]$ to be a
sublocus of a larger scheme $X[A^{\prime},B^{\prime}]$, where the embedding is
given by (3.1.2). This operation of placing a family within a larger family is
exactly what allows us to make all appropriate modifications to $X_{0}$ in the
expanded degeneration setup. We may describe blow-ups of $\mathcal{X}$ by
adding edges to the re-scaled triangle which create vertices by intersecting
at these new integral points.

#### Assigning tropical $\Delta$-multiplicity.

The next step is to assign a tropical $\Delta$-multiplicity to each edge of
$\operatorname{trop}(\mathcal{X})$. By definition, this must be greater or
equal to 1. It does not affect the $\Delta$-type of the vertices attached to
this edge. Let $e$ be an edge of $\Delta$-type $1$ and multiplicity $k$
attached to the vertex $(a,b,0)$ of the triangle (where the coordinates are
given in the 3-dimensional cone). Let $b^{\prime}$ be the smallest integer
such that $b^{\prime}>b$ and such that $(a^{\prime},b^{\prime},0)$ has an edge
attached for some $a^{\prime}$. Let $b^{\prime\prime}$ be the largest integer
such that $b>b^{\prime\prime}$ and such that
$(a^{\prime\prime},b^{\prime\prime},0)$ has an edge attached for some
$a^{\prime\prime}$. We remark here that the $\Delta$-multiplicity $k$ of $e$
does not need to equal the number of integral points up until the next edge,
i.e. $k$ is independent of the values $b^{\prime}-b$ or $b-b^{\prime\prime}$.
A similar statement can be made for $e$ of $\Delta$-type $2$.

Note that whereas on the level of schemes, $\Delta$-multiplicity makes an
important difference, on the level of stacks varying the $\Delta$-multiplicity
preserves the equivalence class of the fibre $\mathcal{X}$. Even though here
we work with stacks, it is still important to consider the underlying schemes
in order to understand what smoothings exist.

#### Sliding the edges.

Where edges have tropical $\Delta$-multiplicity greater than 1, we can slide
out as many copies of this edge as the number of the multiplicity, as long as
the new configuration is compatible with the unbroken condition. These edges
cannot cross another parallel edge. Keeping the notation as above, let
$e^{\prime}$ denote the edge attached to $(a^{\prime},b^{\prime},0)$ and let
$e^{\prime\prime}$ be the edge attached to
$(a^{\prime\prime},b^{\prime\prime},0)$. These may be of type 1 or 2. Sliding
edges from $e$ can be described formally as defining a new unbroken
configuration where there are $r$ edges between $e^{\prime}$ and
$e^{\prime\prime}$, including $e$, and the tropical $\Delta$-multiplicities of
these $r$ edges should sum to $k$. The unbroken condition imposes some
restrictions on how the edges can slide. The situation is similar if we start
with an edge $e$ of type 2, or we may simultaneously slide edges of type 1 and
2 which are attached to the same point of $Y_{1}\cap Y_{2}$.

Wherever one of these new edges which slides out from $e$ intersects another
edge, it creates a new vertex, which corresponds to a new bubble on the
geometric side.

#### Criterion for smoothability.

Let $\mathcal{X}$ and $B_{0}$ as above. Let $v_{0}$ be the vertex in
$\operatorname{trop}(\mathcal{X})$ corresponding to $B_{0}$ and let
$\\{e_{j}\\}$ be the edges of $\Delta$-type 1 and 2 going through this vertex.
There are at most two and each one has a given multiplicity. Let
$\\{v_{i}\\}_{i\geq 1}$ be the collection of all vertices resulting from
intersections which arise when the copies of the edges $e_{i}$ are allowed to
slide to an unbroken configuration in the way described above. Moreover, we
include in the collection $\\{v_{i}\\}_{i\geq 1}$ only vertices which share an
edge with $v_{0}$. Finally, let $\\{B_{i}\\}_{i\geq 1}$ be the geometric
bubbles corresponding to these vertices.

######  Proposition 5.19.

There is a smoothing from $B_{0}$ to each of the $B_{i}$ and there exist no
other smoothings from $B_{0}$. Therefore, a subcheme $Z$ supported in $B_{0}$
is either fixed or tends towards the interior of a component $B_{i}$.

###### Proof.

In étale local coordinates, the bubble $B_{0}$ is given by the equations (3.1)
describing the family $X[A,B]$, as well as some additional equations which fix
certain variables in order to determine $B_{0}$ within the family. We consider
the nonzero equations defining $B_{0}$. Each of these nonzero equations
necessarily contains coordinates corresponding to some $\mathbb{P}^{1}$
associated to a $\Delta$-component. The variables in these equations may vary
as long as both sides of the equation vary together so that the equalities are
preserved. Note that if the base codimension is greater or equal to one, then
varying the values on both sides of the equation preserves the equivalence
class of $\mathcal{X}$ in $\mathfrak{X}^{\prime}$. The equivalence class
changes only when both sides go to zero, in which case the new equivalence
class this gives is in the closure of the first one. A bubble $B^{\prime}$ has
a smoothing from $B_{0}$ if and only if its equations can be obtained from
those of $B_{0}$ by letting both sides of some nonzero equations of $B_{0}$
tend to zero. Therefore, all equations which define $B_{0}$ must hold for
$B^{\prime}$, where some nonzero equations of $B_{0}$ may go to zero for
$B^{\prime}$. In particular, $B^{\prime}$ cannot be determined by a nonzero
equation which does not hold for $B_{0}$. Equivalently, smoothings do not
allow us to enter $\Delta$-components in which we were not before.

This is exactly what the set of bubbles $\\{B_{i}\\}_{i\geq 1}$ defined above
gives us. They correspond to vertices $v_{i}$ created by sliding out and
intersecting copies of the edges of $\Delta$-types 1 and 2 passing through
$v_{0}$. Since each vertex $v_{i},i\geq 1$, must share an edge with $v_{0}$,
the set of $\Delta_{j}^{(k)}$-components which are equal to the corresponding
bubble $B_{i}$ must be a subset of the set of $\Delta_{j}^{(k)}$-components
which are equal to $B_{0}$. It follows that no nonzero equation can hold for a
bubble in $\\{B_{i}\\}_{i\geq 1}$ which does not hold for $B_{0}$. Any vertex
which is not created in this way or does not share an edge with $v_{0}$ will
either correspond to a bubble equal to a $\Delta_{j}^{(k)}$-component which is
not equal to $B_{0}$ and will therefore be defined by some nonzero equation
which does not hold for $B_{0}$; or will correspond to a component which is
not equal to any $\Delta_{j}^{(k)}$, which means it must be a $Y_{i}$
component, and therefore one of the equations $x=0$, $y=0$ or $z=0$ which held
for $B_{0}$ does not hold for it. ∎

_Examples of smoothings._ We can consider a triangle with added edges and
vertices defining some $\operatorname{trop}(\mathcal{X})$ like that given on
the left of Figure 6. Here, we have chosen some vertex which we label $v_{0}$
and we label the edges going through this vertex by $e_{1}$ and $e_{2}$. If,
for example, the edge $e_{1}$ has tropical $\Delta$-multiplicity greater than
1, then we may slide out an edge $e^{\prime}_{1}$ as shown in blue on the
right hand side of Figure 6. The tropical $\Delta$-multiplicities of $e_{1}$
and $e^{\prime}_{1}$ in the second picture should add up to the tropical
$\Delta$-multiplicity of $e_{1}$ in the first picture. The new vertices
created by intersecting the edge $e^{\prime}_{1}$ with the existing edges of
$\operatorname{trop}(\mathcal{X})$ are drawn in blue. We can then see that
there exists a smoothing from $v_{0}$ to $v_{1}$, as they share an edge, but
no smoothings from $v_{0}$ to $u$ or $w$.

$v_{0}$$e_{1}$$e_{2}$$Y_{1}$$Y_{3}$$Y_{2}$

$v_{0}$$e_{1}$$e_{2}$$e^{\prime}_{1}$$v_{1}$$w$$u$$Y_{1}$$Y_{3}$$Y_{2}$

Figure 6: Tropical representation of smoothings.

##  6 Hilbert schemes of points on K3 surfaces and MMP

Up until now, we have only been considering the property that $X\to C$ is
semistable. In this section, we will discuss what properties our constructions
have if we start with some added assumptions. We will want to show that the
proper stacks we construct are semistable themselves, in the following sense.

######  Definition 6.1.

Let $\mathfrak{Y}$ be a Deligne-Mumford stack which is flat and locally of
finite type over $C$. For any $C$-scheme $S$ with a (non-constant) étale
morphism $\xi\colon S\to C$ and any $Y\in\operatorname{Ob}(\mathfrak{Y}_{C})$
over $C$, we denote the composition map $f\colon\xi^{*}Y\to C$. We say that
$\mathfrak{Y}\to C$ is _semistable_ if every such composition map $f$ is
semistable.

Moreover, if $K_{X}+(X_{0})_{\operatorname{red}}$ is semi-ample, i.e. $X\to C$
is a _good semistable model_ , we would like to show that the proper stacks
constructed in previous sections also satisfy this property, which we extend
to stacks in the following way.

######  Definition 6.2.

Let $\mathfrak{Y}$ be a Deligne-Mumford stack which is flat and locally of
finite type over $C$. For any $C$-scheme $S$ with a (non-constant) étale
morphism $\xi\colon S\to C$ and any $Y\in\operatorname{Ob}(\mathfrak{Y}_{C})$
over $C$, let $W\coloneqq\xi^{*}Y$ and denote by $f\colon W\to C$ the
composition map. We say that $\mathfrak{Y}\to C$ is a _good minimal model_ if
for every such $W$ and $f$, the divisor $K_{W/C}+f^{-1}(0)$ is semi-ample.

In particular, we will want to consider the case where $X\to C$ is a good type
III degeneration of K3 surfaces in the sense of Kulikov [Kul77]. This means
that $\pi\colon X\to C$ is semistable, that its general fibres are smooth K3
surfaces and that the dual complex of its central fibre $X_{0}$ is a
triangulated sphere. Moreover, there exists a relative logarithmic 2-form
$\omega_{\pi}\in H^{0}(X,\Omega^{2}_{X/C}(\log X))$ such that
$\wedge^{n}\omega_{\pi}\in H^{0}(X,K_{X/C})$ is nowhere vanishing. In this
case, we will show that such a 2-form on $X\to C$ induces a nowhere degenerate
logarithmic 2-form on the proper stacks we constructed.

Most of the results in this section follow easily from the results of
Gulbrandsen, Halle, Hulek and Zhang in [GHHZ21]. For clarity, however, we give
certain of the relevant proofs again here, where the details are slightly
different for our situation.

###  6.1 Good semistable and dlt minimal models