# The stability manifold of local orbifold elliptic quotients

Franco Rota  Department of Mathematics, University of Utah, Salt Lake City, UT
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate the stability manifold of local models of
orbifold quotients of elliptic curves. In particular, we describe a component
of the stability manifold which maps as a covering space onto the universal
unfolding space of the mirror singularity. The construction requires a
detailed description of the McKay correspondence [9] for $A_{N}$ surface
singularities and a study of wall-crossing phenomena.

###### 2010 Mathematics Subject Classification:

18E30; 14H45, 14J33

###### Contents

  1. 1 Introduction
  2. 2 Stability conditions
  3. 3 Elliptic root systems
  4. 4 Triangulated categories associated to local elliptic quotients
  5. 5 Stability conditions on $\mathcal{D}$
  6. 6 Wall-crossing

##  1\. Introduction

The space of stability conditions on a triangulated category $\mathcal{D}$ was
introduced by Bridgeland in [5], following work of Douglas on $\Pi$-stability
in string theory [10]. Bridgeland shows that the set of these stability
conditions is a complex manifold $\operatorname{Stab}(\mathcal{D})$ [5],
equipped with a local isomorphism

$\operatorname{Stab}(\mathcal{D})\to\operatorname{Hom}(K(\mathcal{D}),\mathbb{C}).$

The stability manifold is fully understood in the case when $\mathcal{D}$ is
the derived category of coherent sheaves on a smooth projective curve (see [5]
for the elliptic curve, [21] for curves of positive genus, and [3], [25] for
the projective line). In the case that $E$ is an elliptic curve, the stability
manifold acquires a mirror-symmetric interpretation, in fact, it can be
expressed as a $\mathbb{C}^{*}$-bundle over the modular curve [7].

In this work, we find a similar description for the stability manifold
associated with the orbifold quotient of an elliptic curve by a group of
automorphisms. Every such quotient has $\mathbb{P}^{1}$ as coarse moduli
space, and it has $p_{1},...,p_{n}$ orbifold points with stabilizers
$\mu_{r_{i}}$ at the point $p_{i}$, we denote it by
$\mathbb{P}^{1}_{r_{1},...,r_{n}}$. Over the field of complex numbers, there
are only two possibilities for special automorphism groups, namely
$\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/6\mathbb{Z}$. These give rise to
three possible quotients: $\mathbb{P}^{1}_{3,3,3}$, $\mathbb{P}^{1}_{4,4,2}$
and $\mathbb{P}^{1}_{6,3,2}$.

The mirror partners of these quotients are _simple elliptic singularities_
[19],[24], described by the following equations:

$\begin{split}E_{6}^{(1,1)}\colon x^{3}+y^{3}+z^{3}+\lambda xyz;\\\
E_{7}^{(1,1)}\colon x^{4}+y^{4}+z^{2}+\lambda xyz;\\\ E_{8}^{(1,1)}\colon
x^{6}+y^{3}+z^{2}+\lambda xyz.\end{split}$

Saito introduces the _universal unfolding spaces_ for these singularities, and
observes that its geometry is regulated by elliptic root systems [26]. The
main result in this paper expresses a relation between the stability manifold
of the orbifold quotients and the universal unfolding of the mirror
singularity.

Rather than the orbifold themselves, we consider their local models. This has
two main advantages: the structure of an elliptic root system is more evident,
and one can use the McKay correspondence to compare the local orbifold to a
smooth surface. From this point of view, local orbifold elliptic quotients
represent an analog of Kleinian singularities.

### Summary of the results

Let $X$ be one of the orbifold elliptic quotients above, embedded as the zero
section in the space $\operatorname{Tot}(\omega_{X})$ of its total canonical
bundle, and let $\mathcal{D}$ be the triangulated category generated by
sheaves supported on $X$, it is a K3-category. Consider $K(X)\simeq
K(\mathcal{D})$, and the symmetric bilinear form $\chi\colon
K(\mathcal{D})\times K(\mathcal{D})\to\mathbb{Z}$ defined as

$\chi(E,F)\coloneqq\sum\limits_{i=0}^{\infty}(-1)^{i}\dim_{\mathbb{C}}\operatorname{Hom}_{\mathcal{D}}(E,F[i])$

called the Euler form. We show that there is an identification of
$K(\mathcal{D})$ with the root lattice of an elliptic root system, which
respects the Euler form.

Then, the Weyl group $W$ acts on
$\operatorname{Hom}(K(\mathcal{D}),\mathbb{C})$ and defines a set of regular
orbits $X_{reg}$. We study a fundamental domain $D$ for this action, and find
a region $U$ in the stability manifold which is homeomorphic to $D$.

A key step in this construction is the McKay correspondence [9]: the
equivalence of categories between $D^{b}(\operatorname{Tot}(\omega_{X}))$ and
the minimal resolution $S$ of its coarse space induces an equivalence between
$\mathcal{D}$ and the triangulated category $\mathcal{D}^{\prime}$ generated
by sheaves supported on the pull-back of the zero section to $S$. We define
$\mathcal{A}\subset\mathcal{D}$ as the pull-back of the standard heart
$\operatorname{Coh}(S)\cap\mathcal{D}^{\prime}\subset\mathcal{D}^{\prime}$,
and observe that $(Z,\mathcal{A})$ is a stability condition for all $Z\in D$.

We show that the connected component
$\operatorname{Stab}^{\circ}(\mathcal{D})$ containing the region $U$ coincides
with the a priori smaller region

(1)
$\operatorname{Stab}^{\dagger}(\mathcal{D})\coloneqq\left\\{\sigma=(Z,\mathcal{P})\in\operatorname{Stab}^{\circ}(\mathcal{D})\quad\middle|\quad(\ast)\colon\operatorname{Im}\frac{Z(b)}{Z(a)}>0\right\\}.$

To prove that
$\operatorname{Stab}^{\circ}(\mathcal{D})=\operatorname{Stab}^{\dagger}(\mathcal{D})$,
we investigate wall-crossing for some specific classes in $K(\mathcal{D})$. As
a result we show Theorem 6.4:

######  Theorem 1.1.

Let $\alpha$ be a root in the elliptic root lattice $K(\mathcal{D})$. Let
$\sigma\in\operatorname{Stab}^{\circ}(\mathcal{D})$ be generic with respect to
$\alpha$. Then, there exists a $\sigma$-stable object $E$ of class $\alpha$.
The object $E$ is rigid if $\alpha$ is a real root, and it varies in a family
if $\alpha$ is imaginary.

Seidel and Thomas [28] define autoequivalences
$\Phi_{S}\in\operatorname{Aut}(\mathcal{D})$ associated to spherical objects
called spherical twists, we denote by $\operatorname{Br}(\mathcal{D})$ the
subgroup of $\operatorname{Aut}(\mathcal{D})$ they generate. The action of
$\operatorname{Br}(\mathcal{D})$ preserves the component
$\operatorname{Stab}^{\dagger}(\mathcal{D})$, and $U$ is a fundamental domain
for this action.

The main result of this paper is the following theorem. It extends results by
Bridgeland and Thomas [8], [30] on Kleinian singularities, and of Ikeda [15]
for arbitrary root systems of symmetric Kac-Moody Lie algebras. Moreover, it
represents a partial answer to Conjecture 1.3 in [29].

######  Theorem 1.2.

There is a covering map

$\bar{\pi}\colon\operatorname{Stab}^{\dagger}(\mathcal{D})\to
X_{reg}/\tilde{W},$

and the group $\mathbb{Z}[2]\times\operatorname{Br}(\mathcal{D})$ acts as
group of deck transformations.

Let
$\operatorname{Aut}^{\dagger}(\mathcal{D})\subset\operatorname{Aut}(\mathcal{D})$
be the subgroup of autoequivalence preserving the region
$\operatorname{Stab}^{\dagger}(\mathcal{D})$. Write
$\operatorname{Aut}^{\dagger}_{*}(\mathcal{D})$ for the quotient of
$\operatorname{Aut}^{\dagger}(\mathcal{D})$ by the subgroup of
autoequivalences which act trivially on
$\operatorname{Stab}^{\dagger}(\mathcal{D})$.

######  Corollary 1.3.

There is an isomorphism

$\operatorname{Aut}^{\dagger}_{*}(\mathcal{D})\simeq\mathbb{Z}[1]\times\left(\operatorname{Br}(\mathcal{D})\rtimes\operatorname{Aut}(\Gamma)\right),$

Where $Aut(\Gamma)$ acts on $\operatorname{Br}(\mathcal{D})$ by permuting the
generators.

### Remarks and further problems

  1. (i)

from the point of view of representation theory, the categories $\mathcal{D}$
discussed here are equivalent to the CY-2 completions of Ringel’s canonical
algebras (see [29]);

  2. (ii)

The space $X_{reg}/\tilde{W}$ in Theorem 1.2 is the universal unfolding of the
corresponding elliptic singularity. In this sense, Theorem 1.2 is a mirror-
symmetric result;

  3. (iii)

The automorphism group of a general elliptic curve $E$ is generated by its
involution $\iota$. The quotient $[E/\iota]$ has the form
$\mathbb{P}^{1}_{2,2,2,2}$: Theorems 1.1 and 1.2 continue to hold in this case
with identical proofs. However, a mirror-symmetric interpretation seems less
clear.

As in [6], [8], we expect the following properties:

######  Conjecture 1.4.

  1. (i)

The space $\operatorname{Stab}(\mathcal{D})$ is connected, so that
$\operatorname{Stab}(\mathcal{D})=\operatorname{Stab}^{\circ}(\mathcal{D})$;

  2. (ii)

the space $\operatorname{Stab}(\mathcal{D})$ is simply connected. This would
also show that $\pi_{1}(X_{reg}/\tilde{W})$ is isomorphic to
$\mathbb{Z}[2]\times\operatorname{Br}(\mathcal{D})$.

See [15] and references therein for progress on Conjecture 1.4 in related
frameworks.

### Conventions

We work over the field $\mathbb{C}$ of complex numbers. All abelian and
triangulated categories are assumed to be $\mathbb{C}$-linear. Given a graph
$\Gamma$, we write $\lvert\Gamma\rvert$ to denote the set of its vertices.

### Acknowledgements

I wish to thank my doctoral advisor, Aaron Bertram, for his guidance and
enthusiasm. I am grateful to Bronson Lim and Huachen Chen for the fruitful
discussions on this topic. I thank Arend Bayer for his helpful comments on a
preliminary version of this work, and Michael Wemyss for discussing the ideas
around Lemma 5.13 with the author.

##  2\. Stability conditions

Stability conditions on triangulated categories were first introduced by
Bridgeland and were inspired by work of Douglas on string theory (see [5] and
references therein). We recall here the definition and basic properties of
stability conditions and the stability manifold. We refer the interested
reader to the early work of Bridgeland [5], [6] and to the surveys [13], [22].

In what follows, $\mathcal{D}$ is a triangulated category, with Grothendieck
group $K(\mathcal{D})$.

######  Definition 2.1.

A _slicing_ of $\mathcal{D}$ is a collection
$\mathcal{P}=\\{\mathcal{P}(\phi)\\}_{\phi\in\mathbb{R}}$ of full additive
subcategories of $\mathcal{D}$ satisfying the following properties:

  1. (i)

$\operatorname{Hom}(\mathcal{P}(\phi_{1}),\mathcal{P}(\phi_{2}))=0$ for
$\phi_{1}<\phi_{2}$;

  2. (ii)

for all $E\in\mathcal{D}$ there are real numbers $\phi_{1}>...>\phi_{m}$,
objects $E_{i}\in\mathcal{D}$ and a collection of triangles

${0=E_{0}}$${E_{1}}$${E_{2}}$${...}$${E_{m-1}}$${E_{m}=E}$${A_{1}}$${A_{2}}$${A_{m}}$

where $A_{i}\in\mathcal{P}(\phi_{i})$;

  3. (iii)

$\mathcal{P}(\phi)[1]=\mathcal{P}(\phi+1)$.

The extremes $\phi_{1}$ and $\phi_{m}$ are denoted $\phi^{+}(E)$ and
$\phi^{-}(E)$ respectively. Given a slicing $\mathcal{P}$, for
$\alpha\leq\beta\in\mathbb{R}$ we denote by $\mathcal{P}((\alpha,\beta))$ the
extension closure of the subcategories
$\\{\mathcal{P}(\phi)\,\colon\,\phi\in(\alpha,\beta)\\}$ (similar definitions
work for other intervals in $\mathbb{R}$).

######  Definition 2.2.

A _stability condition_ on $\mathcal{D}$ is a pair $\sigma=(Z,\mathcal{P})$
where:

  1. (i)

$\mathcal{P}$ is a slicing of $\mathcal{D}$;

  2. (ii)

$Z\colon K(\mathcal{D})\to\mathbb{C}$ is an additive homomorphism called the
_central charge_ ;

and they satisfy the following properties:

  1. (1)

For any non-zero $E\in\mathcal{P}(\phi)$,

$Z([E])\in\mathbb{R}_{>0}\cdot e^{i\pi\phi};$

  2. (2)

(Support property) Fix any norm $\lVert\cdot\rVert$ on $K(\mathcal{D})$. Then
we require

$C_{\sigma}\coloneqq\inf\left\\{\dfrac{\lvert
Z([E])\rvert}{\lVert[E]\rVert}\,\colon\,0\neq
E\in\mathcal{P}(\phi),\,\phi\in\mathbb{R}\right\\}>0$

Given a stability condition $\sigma=(Z,\mathcal{P})$, we’ll refer to
$\mathcal{A}_{\sigma}\coloneqq\mathcal{P}((0,1])$ as to the _heart_ associated
to $\sigma$. In fact, $\mathcal{P}((\alpha,\alpha+1])$ is always the heart of
a bounded $t$-structure for all $\alpha\in\mathbb{R}$, and it’s an abelian
category.

If $E\in\mathcal{P}((\alpha,\alpha+1])$ for some $\alpha\in\mathbb{R}$, then
we say that $E$ has _phase_ $\phi$ if $Z([E])\in\mathbb{R}_{>0}\cdot
e^{i\pi\phi}$, for $\phi\in(\alpha,\alpha+1]$. The nonzero objects of
$\mathcal{P}(\phi)$ are said to be $\sigma$_-semistable_ of phase $\phi$, and
the simple objects of $\mathcal{P}(\phi)$ are said to be $\sigma$_-stable_.

For the general theory about bounded $t$-structures, we refer the reader to
[4], here we only recall the following lemma, which will be useful in what
follows.

######  Lemma 2.3.

Let $\mathcal{A},\mathcal{B}\subset D$ be hearts of bounded t-structures on a
triangulated category $D$. If $\mathcal{A}\subset\mathcal{B}$, then
$\mathcal{A}=\mathcal{B}$.

###### Proof.

A consequence of the definition of bounded $t$-structure. ∎

######  Remark 2.4 ([5, Prop. 5.3]).

When one wants to construct stability conditions it is often easier to use an
alternative definition. One can define a stability condition to be
$\sigma=(Z,\mathcal{A})$ where $\mathcal{A}$ is the heart of a bounded
$t$-structure and $Z$ is a stability function with the Harder-Narasimhan and
support property. A _stability function_ is a linear map $Z\colon
K(\mathcal{A})\to\mathbb{C}$ such that any non-zero $E\in\mathcal{A}$,
satisfies $Z([E])\in\mathbb{R}_{>0}\cdot e^{i\pi\phi}$ with $\phi\in(0,1]$.
Then one defines $\phi$ to be the phase of $E$, and declares $E$ to be
$\sigma$-(semi)stable if for all non-zero subobjects $F\in\mathcal{A}$ of $E$,
$\phi(F)<(\leq)\phi(E)$. We say that $Z$ satisfies the HN property if for
every $E\in\mathcal{A}$ there is a unique filtration

$0=E_{0}\subset E_{1}\subset...\subset E_{n-1}\subset E_{n}=E$

such that the quotients $E_{i}/E_{i-1}$ are $\sigma$-semistable of phases
$\phi_{i}=\phi(E_{i}/E_{i-1})$, $\phi_{1}>\phi_{2}>...>\phi_{n}$. The support
property is the same as in Definition 2.2.

The following proposition is a useful tool to check the Harder-Narasimhan
property:

######  Proposition 2.5 ([22, Prop. 4.10]).

Suppose $\mathcal{A}$ is an abelian category, and $Z\colon
K(\mathcal{A})\to\mathbb{C}$ is a stability function. If

  1. (i)

the category $\mathcal{A}$ is noetherian, and

  2. (ii)

the image of $\operatorname{Im}Z$ is discrete in $\mathbb{R}$,

then $Z$ has the Harder-Narasimhan property.

###  2.1. The Stability manifold

Let $\operatorname{Stab}(\mathcal{D})$ denote the set of stability conditions
on $\mathcal{D}$. In [5, Sec. 6], Bridgeland shows that the function

(2) $f(\sigma,\tau)=\sup_{0\neq
E\in\mathcal{D}}\\{\lvert\phi^{+}_{\sigma}(E)-\phi^{+}_{\tau}(E)\rvert,\lvert\phi^{-}_{\sigma}(E)-\phi^{-}_{\tau}(E)\rvert\\}$

determines a generalized metric on $\operatorname{Stab}(\mathcal{D})$ which
makes it into a topological space. Moreover,
$\operatorname{Stab}(\mathcal{D})$ has a rich geometric structure. This is a
consequence of the following result:

######  Theorem 2.6 ([5, Thm. 1.2]).

The central charge map
$\pi\colon\operatorname{Stab}(\mathcal{D})\to\operatorname{Hom}(K(\mathcal{D}),\mathbb{C})$
given by $(Z,\mathcal{P})\mapsto Z$ is a local homeomorphism. In particular,
$\operatorname{Stab}(\mathcal{D})$ is a complex manifold of dimension
$\operatorname{rk}(K(\mathcal{D}))$.

A part of this work will be dedicated to the study of the map $\pi$. This will
require the following lemma.

######  Lemma 2.7 ([5, Lemma 6.4]).

Let $\sigma$, $\tau\in\operatorname{Stab}(\mathcal{D})$ be stability
conditions with $\pi(\sigma)=\pi(\tau)$. If $f(\sigma,\tau)<1$, then
$\sigma=\tau$.

###  2.2. Torsion pairs and tilts of abelian categories

Next, we recall the definition of a _tilt_ of an abelian category
$\mathcal{A}$, which is a technique to produce new abelian subcategories of
$D^{b}(\mathcal{A})$. Indeed, the tilt of a heart of a bounded t-structure is
a new heart in $D^{b}(\mathcal{A})$ [12].

######  Definition 2.8.

Let $\mathcal{A}$ be an abelian category. A _torsion pair_ (or _torsion
theory_) for $\mathcal{A}$ is a pair of full subcategories
$(\mathcal{T},\mathcal{F})$ such that:

  1. (i)

$\text{Hom}\left(\mathcal{T},\mathcal{F}\right)=0$;

  2. (ii)

for any $E\in\mathcal{A}$ there exists a short exact sequence

$0\to E^{\prime}\to E\to E^{\prime\prime}\to 0$

where $E^{\prime}\in\mathcal{T}$ and $E^{\prime\prime}\in\mathcal{F}$.

Given a torsion pair $(\mathcal{T},\mathcal{F})$ on an abelian category
$\mathcal{A}$, we define
$\mathcal{A}^{\sharp}=\langle\mathcal{F}[1],\mathcal{T}\rangle$ to be the
smallest full subcategory of $D^{b}(\mathcal{A})$ containing $\mathcal{F}[1]$
and $\mathcal{T}$ closed under extensions. $\mathcal{A}^{\sharp}$ is called
the _tilt_ of $\mathcal{A}$ along the torsion pair
$(\mathcal{T},\mathcal{F})$. Sometimes we will also refer to
$\mathcal{A}^{\sharp}[-1]=\langle\mathcal{F},\mathcal{T}[-1]\rangle$ as to the
tilt, but no confusion should arise.

##  3\. Elliptic root systems

This section is a brief summary of the theory of Elliptic root systems,
developed by Saito in [26] and [27]. Some of the explicit computations
presented here are carried out in [29] and [16].

######  Definition 3.1.

Let $F$ be a real vector space of rank $l+2$, equipped with a positive
semidefinite symmetric bilinear form $I\colon F\times F\to F$, whose radical
$\operatorname{rad}I$ has rank 2. An _elliptic root system adapted to_ $F$ is
the datum of a set $R$ of non-isotropic elements of $F$, such that

  1. (1)

the additive group generated by $R$, denoted $Q(R)$, is a full sublattice of
$F$. That is, the embedding $Q(R)\subset F$ induces an isomorphism
$Q(R)_{\mathbb{R}}\simeq F$;

  2. (2)

the symmetric bilinear form $I\colon R\times R\to\mathbb{Z}$;

  3. (3)

the group $W$ generated by
$\\{w_{\alpha}\in\operatorname{Aut}(F,I)\,|\,\alpha\in R\\}$, where

$w_{\alpha}(x)=x-I(x,\alpha)\alpha\,\text{ for all }\,x\in F$

preserves $R$;

  4. (4)

if $R=R_{1}\cup R_{2}$ with $R_{1}\perp R_{2}$, then either $R_{1}$ or $R_{2}$
is empty.

######  Definition 3.2.

An elliptic root system $R$ is said to be _oriented_ if $\operatorname{rad}I$
is oriented. An _admissible frame_ of $\operatorname{rad}I$ is an oriented
basis $(a,b)$ of $\operatorname{rad}I$ such that
$Q(R)\cap\operatorname{rad}I\simeq\mathbb{Z}a\oplus\mathbb{Z}b$. Denote by $G$
the subspace $\mathbb{R}a\subset F$. In this case, we refer to the pair
$(R,G)$ as to a _marked_ elliptic affine root system. We refer to $a$ as to a
_signed marking_ of $R$.

From now on, we fix a marked root system $(R,G)$ with a signed marking $a$.

Pick generators $\alpha_{-1},\alpha_{0},\alpha_{1},...,\alpha_{l}$ of $Q(R)$
so that

$F=G\oplus L$

where $L\coloneqq\oplus_{i=0}^{l}\mathbb{R}\alpha_{i}$. The image of $R$ under
the projection $p\colon F\to F/G$ is an affine root system, which will be
denoted $R_{a}$. Similarly, the image of $R$ under the quotient $F\to
F/\operatorname{rad}I$ is a finite root system $R_{f}$.

######  Proposition 3.3 ([26]).

The root system $R$ is given by

$R=\\{\alpha_{f}+mb+na\,|\,\alpha_{f}\in R_{f},m,n\in Z\\}.$

######  Definition 3.4.

The elements of $R$ are also called the real roots of the root system. We
define the set $\Delta_{im}$ of imaginary roots of $R$ as

$\Delta_{im}=\\{mb+na\,|\,m,n\in\mathbb{Z}\\}.$

###  3.1. The Dynkin graph

To a marked elliptic affine root system $(R,G)$ one can associate a diagram
$\Gamma_{R,G}$ called the _Dynkin graph of_ $(R,G)$. We omit the general
construction given in [27], but we recall some of the properties of
$\Gamma_{R,G}$ which will be useful in what follows.

  1. (i)

The set of vertices $\lvert\Gamma_{R,G}\rvert$ is
$\\{\alpha_{-1},\alpha_{0},...,\alpha_{l}\\}$;

  2. (ii)

two vertices $\alpha,\beta\in\lvert\Gamma_{R,G}\rvert$ are connected following
the rule:

${\alpha}$${\beta}$${\circ}$${\circ}$${\mbox{ if }\ \
I(\alpha,\beta)=0;}$${\circ}$${\circ}$${\mbox{ if }\ \
I(\alpha,\beta)=-1;}$${\circ}$${\circ}$${\mbox{ if }\ \ I(\alpha,\beta)=2.}$

######  Example 3.5.

Our main interest lies in the root systems $E_{6}^{(1,1)}$, $E_{7}^{(1,1)}$
and $E_{8}^{(1,1)}$, whose diagrams are:

${\circ}$${\circ}$${\circ}$${E_{6}^{(1,1)}}$${\circ}$${\circ}$${\circ}$${\circ}$${\circ\par}$${\circ}$${\circ}$${\circ}$${\circ}$${E_{7}^{(1,1)}}$${\circ}$${\circ}$${\circ}$${\circ}$${\circ}$${\circ}$${\circ}$${\circ}$${\circ}$${\circ}$${\circ}$${E_{8}^{(1,1)}}$${\circ}$${\circ}$${\circ}$${\circ}$

######  Notation 3.6.

When $\Gamma$ is one of the diagrams above, we introduce a labelling for the
vertices. We use $v_{-1},v_{0}$ to denote the two central vertices. The
diagram $\Gamma^{\prime}$ obtained by deleting $v_{-1},v_{0}$ and all adjacent
vertices is a disjoint union of diagrams of type $A_{r_{i}-1}$, with
$i=1,2,3$. We denote by $v_{(i,j)}$ ($j=1,...,r_{i}-1$) the vertex occupying
the $j-th$ position on the $i-th$ diagram $A_{r_{i}-1}$. We will use this
indexing to label the generators $\alpha_{-1},...,\alpha_{l}$ when it is
convenient.

######  Remark 3.7.

An elliptic affine root system can be viewed as an extension of the
corresponding affine root system. This can be seen by looking at the Dynkin
diagrams: one recovers the affine diagram $\Gamma_{a}$ associated to $R_{a}$
by erasing from $\Gamma_{R,G}$ the vertex $v_{-1}$ and all the edges
connecting $v_{-1}$ to other vertices.

###  3.2. The Weyl group

The projection $p\colon F\to F/G$ induces a homomorphism $p_{*}\colon W\to
W_{a}$ to the affine Weyl group associated to $R_{a}$. Denote by $T$ the
kernel of $p_{*}$.

######  Lemma 3.8 ([26, §1.15]).

The subgroup of $W$ generated by $\\{w_{\alpha_{0}},...,w_{\alpha_{l}}\\}$ is
isomorphic to $W_{a}$, so the sequence

(3) $0\to T\to W\to W_{a}\to 1$

splits into a semi-direct product $W=T\rtimes W_{a}$.

The following elements are relevant to our analysis:

######  Definition 3.9.

For each vertex of $\Gamma_{a}$ we define automorphisms of $F$ as follows:

  1. (1)

$r_{v_{0}}\coloneqq w_{\alpha_{0}}w_{\alpha_{-1}}$;

  2. (2)

$r_{v_{(i,1)}}\coloneqq
w_{\alpha_{(i,1)}}r_{v_{0}}w_{\alpha_{(i,1)}}r_{v_{0}}^{-1}$ for $i=1,2,3$;

  3. (3)

$r_{v_{(i,j)}}\coloneqq
w_{\alpha_{(i,j)}}r_{v_{(i,j-1)}}w_{\alpha_{(i,j)}}r_{v_{(i,j-1)}}^{-1}$ for
$i=1,2,3$, $j=2,...,r_{i}-1$;

######  Lemma 3.10 ([29, Sec. 3]).

For all $v\in\Gamma_{a}$, the automorphism $r_{v}$ belongs to $W$. For
$\beta\in F$, we have

$r_{v}(\beta)=\beta-I(\beta,\alpha_{v})a.$

Moreover, there is a group homomorphism

$\displaystyle\varphi\colon Q(R_{a})$ $\displaystyle\to W$
$\displaystyle\sum_{i=0}^{l}m_{i}\alpha_{i}$
$\displaystyle\mapsto\prod_{i}r_{i}^{m_{i}}$

with kernel generated by $b$. The lattice $Q(R_{f})\simeq\varphi(Q(R_{a}))$ is
isomorphic to $T$, and $\varphi$ induces the inclusion $T\to W$ of the exact
sequence (3).

Next, we recall some aspects of Saito’s construction of the universal
unfolding space of simple elliptic singularities. From now on, fix a marked
elliptic root system $(R,G)$ with an oriented basis $(a,b)$ for
$\operatorname{rad}I$ and keep the notation as above.

######  Definition 3.11.

Up to a linear isomorphism, there is a unique real vector space $\tilde{F}$ of
rank $l+3$ with:

  1. (i)

an inclusion $F\subset\tilde{F}$;

  2. (ii)

a symmetric bilinear form
$\tilde{I}\colon\tilde{F}\times\tilde{F}\to\tilde{F}$ such that
$\tilde{I}_{|F}=I$ and $\operatorname{rad}\tilde{I}=\mathbb{R}a$.

We say $(\tilde{F},\tilde{I})$ is a _hyperbolic extension_ of $(F,I)$. We fix
a basis $\tilde{\lambda}$ of $\tilde{F}/F$ normalized as

$\displaystyle\tilde{I}(\tilde{\lambda},b)=1;$
$\displaystyle\tilde{I}(\tilde{\lambda},\alpha_{i})=0\mbox{ for }i=1,...,l.$

Then, for $\alpha\in R$ and $\gamma\in\tilde{F}$, one defines

$\tilde{w}_{\alpha}\colon\gamma\mapsto\gamma-\tilde{I}(\gamma,\alpha)\alpha$

and

$\tilde{W}\coloneqq\langle\tilde{w}_{\alpha}\,|\,\alpha\in R\rangle.$

Define moreover the map
$\varsigma\colon\gamma\mapsto\gamma-\tilde{I}(\gamma,b)a$. We have the
following description:

######  Lemma 3.12 ([27, §2.7]).

There is a commutative diagram with exact rows and columns:

${K}$${\tilde{T}}$${T}$${K}$${\tilde{W}}$${W}$${W_{a}}$${W_{a}}$

where $K$ is infinite cyclic generated by $\varsigma$, and the rightmost
column is the exact sequence (3).

###  3.3. The regular set

The goal of this section is to introduce domains for actions of the groups
involved in Lemma 3.12. One defines three domains

$\displaystyle\tilde{\mathbb{E}}\coloneqq\\{x\in\operatorname{Hom}(\tilde{F},\mathbb{C})\,|\,x(a)=1,\operatorname{Im}x(b)>0\\};$
$\displaystyle\mathbb{E}\coloneqq\\{x\in\operatorname{Hom}(F,\mathbb{C})\,|\,x(a)=1,\operatorname{Im}x(b)>0\\};$
$\displaystyle\mathbb{H}\coloneqq\\{x\in\operatorname{Hom}(\operatorname{rad}I,\mathbb{C})\,|\,x(a)=1,\operatorname{Im}x(b)>0\\}.$

We define an action of $\tilde{W}$ on $\tilde{\mathbb{E}}$ as:

$(gx)(u)\coloneqq x(g^{-1}u)$

for $x\in\tilde{\mathbb{E}}$ and $g\in\tilde{W}$. Likewise, $W$ acts on
$\mathbb{E}$. The actions above preserve $x_{|\operatorname{rad}I}$, so they
respect the restriction maps $\tilde{\mathbb{E}}\to\mathbb{E}\to\mathbb{H}$.

The element $\tilde{\lambda}\in\tilde{F}/F$ can be viewed as a complex
coordinate for $\tilde{\mathbb{E}}$ over $\mathbb{E}$. By our choice of
$\tilde{\lambda}$, the quantity $\lambda\coloneqq\exp\\{2\pi
i\tilde{\lambda}\\}$ is invariant under the action of
$K=\mathbb{Z}\\{\varsigma\\}$.

Rather than $\tilde{\mathbb{E}}$, it will be convenient to consider
$\tilde{\mathbb{E}}/K$, which is a trivial $\mathbb{C}^{*}$-bundle over
$\mathbb{E}$ with fiber coordinate $\lambda$.

######  Proposition 3.13 ([27, §3, §4]).

The action of $\tilde{W}$ (resp. of $W$) on $\tilde{\mathbb{E}}$ (resp. on
$\mathbb{E}$) is properly discontinuous. Moreover, it is fixed point free on

$\tilde{X}_{reg}\coloneqq\tilde{\mathbb{E}}\setminus\cup_{\alpha\in
R}\tilde{H}_{\alpha},$

where $\tilde{H}_{\alpha}$ is the hyperplane defined by the equation
$x(\alpha)=0$.

######  Definition 3.14.

We denote by $X_{reg}^{N}$ the normalized regular set for $W$, defined as

$X_{reg}^{N}\coloneqq\mathbb{E}\setminus\cup_{\alpha\in R}H_{\alpha}$

where $H_{\alpha}$ is the hyperplane defined by the equation $x(\alpha)=0$.

It is clear from the definitions that $\tilde{H}_{\alpha}=\mathbb{C}\times
H_{\alpha}$ for all $\alpha\in R$, so we have
$\tilde{X}_{reg}\simeq\mathbb{C}\times X_{reg}^{N}$. We write

$X_{reg}\coloneqq\tilde{X}_{reg}/K\simeq\mathbb{C}^{*}\times X_{reg}^{N}.$

There are two group actions on $X_{reg}$ which commute with each other: the
Weyl group $W$ acts by reflections on $X_{reg}^{N}$ and leaves
$\mathbb{C}^{*}$ fixed, while $\mathbb{C}^{*}$ acts on the first factor by
multiplication. The embedding

$X_{reg}\simeq\mathbb{C}^{*}\times
X_{reg}^{N}\longrightarrow\operatorname{Hom}(F,\mathbb{C})$

given by $(t,x)\mapsto tx$ is equivariant with respect to the actions of $W$
and $\mathbb{C}^{*}$. Therefore, we think of
$X_{reg}\subset\operatorname{Hom}(F,\mathbb{C})$.

###  3.4. Fundamental domain

Our goal is now to describe a fundamental domain for the action of $W$ on
$\mathbb{E}$. We introduce the following notation for the tangent spaces of
$\mathbb{E}$ and $\tilde{\mathbb{E}}$ relative to $\mathbb{H}$:

$V_{\mathbb{C}}\coloneqq V\otimes_{\mathbb{R}}\mathbb{C}\,\text{ where
}\,V\coloneqq(F/\operatorname{rad}I)^{*}$
$\tilde{V_{\mathbb{C}}}\coloneqq\tilde{V}\otimes_{\mathbb{R}}\mathbb{C}\,\text{
where }\,\tilde{V}\coloneqq(\tilde{F}/\operatorname{rad}I)^{*}.$

The bilinear forms $I$ and $\tilde{I}$ induce isomorphims

$I^{*}\colon V\xrightarrow{\sim}V^{*}=F/\operatorname{rad}I$
$\tilde{I}^{*}\colon\tilde{V}\xrightarrow{\sim}V^{*}=\tilde{F}/\operatorname{rad}I=F/G$

For $\tau\in\mathbb{H}$, consider moreover the map
$\varphi_{\tau}\colon\operatorname{rad}I\simeq\mathbb{C}$ defined by

$\varphi_{\tau}\colon ua+vb\mapsto u+v\tau.$

Then, one has a family of isomorphisms of complex vector spaces

(4) $\varphi_{\tau}\otimes
I\colon\operatorname{rad}I\otimes_{\mathbb{R}}(F/\operatorname{rad}I)\simeq
V_{\mathbb{C}}.$

######  Lemma 3.15 ([27, §3]).

  1. (i)

$W$ acts preserving fibers $\mathbb{E}_{\tau}$ above a point in
$\tau\in\mathbb{H}$;

  2. (ii)

We have an identification $\mathbb{E}_{\tau}\simeq V_{\mathbb{C}}\simeq
V\oplus\tau V$. The group $W_{a}$ acts on $\tau V$ by reflections, and $T$ is
a finite index subgroup of the real translation lattice $Q(R_{f})\subset V$.

To the affine root system $R_{a}$ we can associate the Weyl alcove

$A_{\mathbb{R}}\coloneqq\\{h\in Q(R_{a})_{\mathbb{R}}^{*}\,|\,h(\alpha_{v})>0\
\ \mbox{ for }v\in\lvert\Gamma_{a}\rvert\\}$

and the Tits cone

$\overline{\mathsf{T}_{\mathbb{R}}(R_{a})}\coloneqq\bigcup\limits_{w\in
W_{a}}w\overline{A_{\mathbb{R}}},$

where $\mathsf{T}_{\mathbb{R}}(R_{a})$ denotes the topological interior of
$\overline{\mathsf{T}_{\mathbb{R}}(R_{a})}$.

######  Remark 3.16.

It is known that $\overline{A_{\mathbb{R}}}$ is a fundamental domain for the
action of $W_{a}$ on $\overline{\mathsf{T}_{\mathbb{R}}(R_{a})}$ [17].

The _complexified Tits cone_ associated to $R_{a}$ is

$\mathsf{T}(R_{a})\coloneqq\\{h\in
Q(R_{a})_{\mathbb{C}}^{*}\,|\,\operatorname{Im}h\in\mathsf{T}_{\mathbb{R}}(R_{a})\\}.$

The complexified Tits cone can be equivalently described as

$\mathsf{T}(R_{a})=\\{h\in Q(R_{a})_{\mathbb{C}}^{*}\,|\,h(b)\in\mathbb{H}\\}$

(see the discussion in [15, Section 2.3]).

Denote by $A\subset V_{\mathbb{C}}$ the complexified Weyl alcove

$A\coloneqq\\{h\in V_{\mathbb{C}}\,|\,\operatorname{Im}(h(\alpha_{v}))>0\mbox{
for }v\in\lvert\Gamma_{a}\rvert\\},$

and let $A_{\tau}$ be its image in $\mathbb{E}_{\tau}$ under the isomophism
(4).

Let $B^{\prime}$ be a hypercube in $V$ which contains the origin and is a
fundamental domain for the action of $T$ on $V$, and define
$B_{\tau}\coloneqq\\{h\in\mathbb{E}_{\tau}\simeq
V_{\mathbb{C}}\,|\,\operatorname{Re}(h)\in B^{\prime}\\}$.

######  Proposition 3.17.

A fundamental domain for the action of $W$ on $\mathbb{E}_{\tau}$ is the
intersection

$D_{\tau}\coloneqq A_{\tau}\cap B_{\tau}.$

A fundamental domain for the action of $W$ on $\mathbb{E}$ is
$D\coloneqq\cup_{\tau}D_{\tau}\simeq D_{\sqrt{-1}}\times\mathbb{H}\subset
X^{N}_{reg}$.

###### Proof.

As a consequence of Prop. 3.13, it is enough to show that for every
$Z\in\mathbb{E}_{\tau}$ there exists an element $w\in W$ such that $w\cdot
Z\in D_{\tau}$. Using the complex structure given in (4), we may write every
$Z\in\mathbb{E}_{\tau}$ as $\operatorname{Re}Z+\tau\operatorname{Im}Z$. As a
consequence of Remark 3.16, there exists an element $w^{\prime}\in W_{a}$ such
that $w^{\prime}\cdot\operatorname{Im}Z\in A_{\tau}$. Then $w^{\prime}\cdot Z$
belongs to $V\oplus iV$. By definition of $B_{\tau}$, there is an element
$r\in T$ such that $\operatorname{Re}(r,w^{\prime})\cdot
Z=\operatorname{Re}w^{\prime}\cdot Z\in B_{\tau}$ and
$\operatorname{Im}(r,w^{\prime})\cdot Z=\operatorname{Im}w^{\prime}\cdot Z\in
A_{\tau}$.

The statement about $\mathbb{E}$ follows, since every $w\in W$ preserves the
fibers $\mathbb{E}_{\tau}$ by Lemma 3.15. ∎

We now describe the boundary of $\overline{D}$ in $X^{N}_{reg}$ in terms of
walls for the action of $W$. For vertices $v\in\lvert\Gamma_{a}\rvert$ we
define walls $W_{v,\pm}\subset\overline{D}$ for the Weyl alcove

$\displaystyle W_{v,+}\coloneqq\\{Z\in
X^{N}_{reg}\cap\overline{D}\,|\,Z(\alpha_{v})\in\mathbb{R}_{>0},\operatorname{Im}Z(\alpha_{u})>0\mbox{
for }u\neq v\\}$ $\displaystyle W_{v,-}\coloneqq\\{Z\in
X^{N}_{reg}\cap\overline{D}\,|\,Z(\alpha_{v})\in\mathbb{R}_{<0},\operatorname{Im}Z(\alpha_{u})>0\mbox{
for }u\neq v\\}$

For vertices $v_{(i,j)}\in\lvert\Gamma_{R}\rvert$, write
$Y^{\prime}_{(i,j),\pm}$ for the faces of the fundamental hypercube
$B^{\prime}$, and let

$Y_{(i,j),\pm}\coloneqq\cup_{\tau}(Y^{\prime}_{(i,j),\pm}\oplus\tau V)\subset
X^{N}_{reg}\cap\overline{D}.$

Then, the boundary of $D$ in $X^{N}_{reg}$ is contained in the union of the
walls $W_{v,\pm}$ and $Y_{(i,j),\pm}$ as $i,j$ vary.

###  3.5. Fundamental group

In this section we describe the fundamental group of
$X_{reg}/W=\tilde{X}_{reg}/\tilde{W}$.

######  Definition 3.18.

Let $R$ be an elliptic root system. The Artin group $G_{W}$ associated with
the Weyl group $W$ is the group generated by
$\\{g_{v},h_{v}\,|\,v\in\lvert\Gamma_{a}\rvert\\}$ with relations

$\displaystyle g_{v}g_{u}=g_{u}g_{v}$ $\displaystyle\;\mbox{ if
}\;I(\alpha_{v},\alpha_{u})=0;$ $\displaystyle
g_{v}g_{u}g_{v}=g_{u}g_{v}g_{u}$ $\displaystyle\;\mbox{ if
}\;I(\alpha_{v},\alpha_{u})=-1;$ $\displaystyle h_{v}h_{u}=h_{u}h_{v}$
$\displaystyle\;\mbox{ for all }\;u,v\in\lvert\Gamma_{a}\rvert;$
$\displaystyle g_{v}h_{u}=h_{u}g_{v}$ $\displaystyle\;\mbox{ if
}\;I(\alpha_{v},\alpha_{u})=0;$ $\displaystyle g_{v}h_{u}g_{v}=h_{u}h_{v}$
$\displaystyle\;\mbox{ if }\;I(\alpha_{v},\alpha_{u})=-1;$

######  Proposition 3.19.

Suppose $R$ is an elliptic root system. Then, the fundamental group of
$X_{reg}/W$ is given by

$\pi_{1}(X_{reg}/W,*)\simeq\mathbb{Z}[\eta]\times G_{W}.$

The path $[\eta]$ corresponds to the $S^{1}$-orbit of $*$ in $\mathbb{C}^{*}$.
The generator $g_{v}$ of $G_{W}$ is given by the path connecting $*$ and
$w_{\alpha_{v}}(*)$ passing through $W_{v,+}$ just once. The generator $h_{v}$
of $G_{W}$ is given by the path connecting $*$ and $r_{v}(*)$ which is
constant in the imaginary part.

###### Proof.

We have $X_{reg}\simeq\mathbb{C}^{*}\times X_{reg}^{N}$ and
$\pi_{1}(\mathbb{C}^{*})\simeq\mathbb{Z}$, so it is enough to show that
$\pi_{1}(X_{reg}^{N}/W)\simeq G_{W}$. By construction, $X_{reg}^{N}$ coincides
with the regular subset of the complexified Tits cone

$\mathsf{T}_{reg}(R_{a})\coloneqq\mathsf{T}(R_{a})\setminus\bigcup\limits_{\alpha\in
R}H_{\alpha},$

where $H_{\alpha}$ is the reflection hyperplane defined by $h(\alpha)=0$.

Then, the result in [32] implies $\pi_{1}(\mathsf{T}_{reg}(R_{a})/W)\simeq
G_{W}$, which concludes the proof. ∎

##  4\. Triangulated categories associated to local elliptic quotients

We are interested in studying orbifold curves obtained from a quotient of an
elliptic curve by a finite subgroup of its automorphism groups. Every elliptic
quotient has $\mathbb{P}^{1}$ as coarse moduli space and orbifold points
$p_{i}$ with stabilizers $\mu_{r_{i}}$. Up to permuting the $p_{i}$’s, there
are only 4 possibilities, namely: $\mathbb{P}^{1}_{2,2,2,2}$,
$\mathbb{P}^{1}_{3,3,3}$, $\mathbb{P}^{1}_{4,4,2}$ and
$\mathbb{P}^{1}_{6,3,2}$. These are denoted respectively $X_{2},X_{3},X_{4}$
and $X_{6}$.

Each $X_{r}$ can be realized as a quotient of an elliptic curve $E_{r}$ by a
subgroup $\mu_{r}$ of its automorphism group:

$X_{r}=\left[{\raisebox{1.99997pt}{$E_{r}$}\left/\raisebox{-1.99997pt}{$\mu_{r}$}\right.}\right].$

From now on, we fix $r$ and denote $X\coloneqq X_{r}$, $E\coloneqq E_{r}$.

Consider the embedding of $X$ in the total space
$\operatorname{Tot}(\omega_{X})=\left[\operatorname{Tot}(\omega_{E})/\mu_{r}\right]$
of its canonical bundle. We have a commutative diagram

${E}$${\operatorname{Tot}(\omega_{E})}$${X}$${\operatorname{Tot}(\omega_{X})\eqqcolon
Y}$$\scriptstyle{\iota}$

######  Definition 4.1.

A triangulated category $\mathbb{T}$ is called a K3-category if the functor
$[2]$ is a Serre functor, i.e. if for any two objects $E,F\in\mathbb{T}$ there
is a natural isomorphism

$\nu_{E,F}\colon\operatorname{Hom}(E,F)\xrightarrow{\sim}\operatorname{Hom}(F,E[2])^{*}.$

Let $\mathcal{D}$ denote the full triangulated subcategory of coherent sheaves
supported on the zero section of $Y$. Then we have:

######  Lemma 4.2.

$\mathcal{D}$ is a K3-category. In particular, the Euler form is symmetric.
Moreover, for any $E,F\in D^{b}(X)$, one has

$\operatorname{Hom}^{\bullet}_{\mathcal{D}}(\iota_{*}E,\iota_{*}F)=\operatorname{Hom}^{\bullet}_{X}(E,F)\oplus\operatorname{Hom}^{\bullet}_{X}(F,E)^{*}[-2].$

In particular,
$\chi_{\mathcal{D}}(\iota_{*}E,\iota_{*}F)=\chi_{X}(E,F)+\chi_{X}(F,E).$

###### Proof.

This is a consequence of [18, Lemma 4.4]. ∎

######  Lemma 4.3.

The map $\iota$ induces an isomorphism of abelian groups $K(X)\simeq
K(\mathcal{D})$.

###### Proof.

Let $X_{n}$ be the $n$-th order neighborhood of $X$ in $Y$. Denote by
$\mathcal{B}$ be the abelian category of sheaves supported on $X$. Then any
$F\in\mathcal{B}$ is an $\mathcal{O}_{X_{n}}$-module for some $n$. Therefore,
$F$ is obtained as a successive extension of $\mathcal{O}_{X}$-modules, and
the map

$\iota_{*}K(X)\to K(\mathcal{B})=K(\mathcal{D})$

is surjective. Let $\pi\colon Y\to X$ denote the projection to the zero
section. Since $R^{i}\pi_{*}=0$ for $i>0$, the functor

$\pi_{*}\colon\mathcal{B}\to\operatorname{Coh}(X)$

is exact. The induced map on $K$-groups is the inverse of $\iota_{*}$. ∎

###  4.1. Exceptional and spherical objects

An object $S\in\mathcal{D}$ is called _spherical_ if
$\operatorname{Hom}^{\bullet}(S,S)\simeq\mathbb{C}\oplus\mathbb{C}[-2]$.
Suppose $S\in\mathcal{D}$ is a spherical object. Given an object
$G\in\mathcal{D}$ we define $\Phi_{S}G$ to be the cone of the evaluation
morphism

$\operatorname{Hom}^{\bullet}(S,G)\otimes S\xrightarrow{ev}G\to\Phi_{S}G.$

Similarly, $\Phi^{-}_{S}G$ is a shift of the cone of the coevaluation map

$\Phi^{-}_{S}G\to
G\xrightarrow{ev^{*}}\operatorname{Hom}^{\bullet}(G,S)^{*}\otimes S$

The operations $\Phi_{S}$, $\Phi^{-}_{S}$ define autoequivalences of
$\mathcal{D}$, called _spherical twists_ [28].

Spherical twists act on $K(\mathcal{D})$ via reflections: if $S$ is a
spherical object, and $[G]\in K(\mathcal{D})$, we have

$w_{S}([G])\coloneqq[\phi_{S}G]=[G]-\chi(S,G)[S].$

######  Lemma 4.4.

[28] Let $S$ be a spherical object of $\mathcal{D}$. Then,

  1. (i)

we have $\Phi_{S}\Phi_{S}^{-}\simeq\operatorname{id}_{\mathcal{D}}$ and
$\Phi_{S}^{-}\Phi_{S}\simeq\operatorname{id}_{\mathcal{D}}$;

  2. (ii)

we have $\Phi_{S}S\simeq S[-1]$;

  3. (iii)

for any spehrical object $S^{\prime}$ such that
$\operatorname{Hom}^{\bullet}(S^{\prime},S)\simeq\mathbb{C}[-1]$, there is an
isomorphism

$\Phi_{S}\Phi_{S^{\prime}}S\simeq S^{\prime}.$

Our next goal is to produce spherical objects in $\mathcal{D}$. To do so, we
use the fact that $D^{b}(X)$ admits exceptional collections:

######  Definition 4.5.

Let $\mathbb{T}$ be a triangulated category. An object $E\in\mathbb{T}$ is
_exceptional_ if

$\operatorname{Hom}^{\bullet}(E,E)=\mathbb{C}.$

An _exceptional collection_ is a sequence of exceptional objects
$E_{1},...,E_{n}$ such that $\operatorname{Hom}^{\bullet}(E_{i},E_{j})=0$ for
$i>j$. We say that an exceptional collection is _full_ if it generates
$\mathbb{T}$, i.e. $\mathbb{T}$ is the smallest triangulated category
containing the $\\{E_{i}\\}$.

######  Proposition 4.6 ([28]).

Suppose $E\in D^{b}(X)$ is exceptional, then $\iota_{*}E$ is a sperical object
in $\mathcal{D}$.

###### Proof.

This is a consequence of Prop. 3.15 in [28]. ∎

The category $\operatorname{Coh}(X)$ admits exceptional simple sheaves
$\mathcal{O}_{p_{i}}\chi^{j}$ for $j=0,...,r_{i}-1$ (see, for example, [11]).
In fact, $D^{b}(X)$ admits several full exceptional collections [23]. Our
attention goes to the following exceptional collection

$\mathbb{F}\coloneqq\left(\mathcal{O}_{p_{1}}\chi^{r_{1}-1},...,\mathcal{O}_{p_{1}}\chi^{1},\mathcal{O}_{p_{2}}\chi^{r_{2}-1},...,\mathcal{O}_{p_{2}}\chi^{1},\mathcal{O}_{p_{3}}\chi^{r_{3}-1},...,\mathcal{O}_{p_{3}}\chi^{1},\mathcal{O},\mathcal{O}(1)\right).$

By Prop. 4.6, pushing forward the objects of $\mathbb{F}$, we obtain a set of
spherical objects:

(5)
$\Pi\coloneqq\left(t_{1}^{r_{1}-1},...,t_{1}^{1},t_{2}^{r_{2}-1},...,t_{2}^{1},t_{3}^{r_{3}-1},...,t_{3}^{1},\iota_{*}\mathcal{O},\iota_{*}\mathcal{O}(1)\right),$

where $t_{i}^{j}\coloneqq\iota_{*}\mathcal{O}_{p_{i}}\chi^{j}$. To streamline
the notation, elements of $\Pi$ will be denoted $S_{m}$, where
$m=-1,0,1,...,\sum_{i}r_{i}-3$, choosing indices so that
$S_{0}\coloneqq\iota_{*}\mathcal{O}$ and
$S_{-1}\coloneqq\iota_{*}\mathcal{O}(1)$. Let $\operatorname{Br}(\mathcal{D})$
be the subgroup of $\operatorname{Aut}(\mathcal{D})$ generated by the
spherical twists $\\{\Phi_{S}\\}$ with $S\in\Pi$.

###  4.2. The root system associated to $\mathcal{D}$

This section revolves around the following proposition:

######  Proposition 4.7.

The set
$R\coloneqq\operatorname{Br}(\mathcal{D})\Pi=\\{[\Phi_{S}]\,|\,S\in\Pi,\Phi\in\operatorname{Br}(\mathcal{D})\\}$
satisfies the axioms of an extended root system adapted to
$\left(K(\mathcal{D})_{\mathbb{R}},\chi_{\mathcal{D}}\right)$ (see Def. 3.1).
Moreover:

  1. (i)

The radical of $I\coloneqq\chi_{\mathcal{D}}$ is generated by the image of
$K(E)$ in $K(\mathcal{D})$ under the push forward along the quotient map
$p\colon E\to X$;

  2. (ii)

Define classes

$a\coloneqq[\iota_{*}\mathcal{O}]-[\iota_{*}\mathcal{O}(1)],$
$b\coloneqq[p_{*}\mathcal{O}_{E_{r}}].$

Then $(a,b)$ is an admissible frame of $\operatorname{rad}I$ and $a$ is a
signed marking for $R$;

  3. (iii)

The Weyl group $W$ is generated by $\\{w_{S}\,|\,S\in\Pi\\}$;

  4. (iv)

the root systems arising from an elliptic orbifold quotient are precisely the
ones described in Example 3.5. The vertices $v_{-1},v_{0}$ correspond to
$\iota_{*}\mathcal{O}(1),\iota_{*}\mathcal{O}$ respectively, and $v_{(i,j)}$
to $t_{i}^{j}$.

###### Proof.

The axioms of an elliptic root system for
$\left(K(\mathcal{D})_{\mathbb{R}},\chi_{\mathcal{D}}\right)$ are verified in
[23]. Observe that the radical $\operatorname{rad}I$ has rank 2, and the
classes

$a=-[\mathcal{O}_{q}]\ \ \mbox{ and }\ \
b=[\mathcal{O}_{X}\oplus\omega_{X}\oplus\omega_{X}^{2}]$

are invariant under twists by $\omega_{X}$, so $a,b\in\operatorname{rad}I$ by
Lemma 4.8. ∎

######  Lemma 4.8.

If $N\in D^{b}(X)$ satisfies $N\otimes\omega_{X}\simeq N$, then
$[\iota_{*}N]\in\operatorname{rad}\chi_{\mathcal{D}}$.

###### Proof.

The classes $[\iota_{*}E]$ for $E\in D^{b}(X)$ generate $K(\mathcal{D})$, and
we have

$\chi_{\mathcal{D}}(\iota_{*}N,\iota_{*}E)=\chi_{X}(N,E)+\chi_{X}(E,N)=\chi_{X}(N,E)-\chi_{X}(E,N\otimes\omega_{X_{r}})=0$

by Lemma 4.2. ∎

Let $\Gamma$ denote the diagram corresponding to $R$, and denote by
$\Gamma_{a}$ the underlying affine Dynkin diagram, obtained by erasing the
vertex $v_{-1}$ and all edges adjacent to it.

######  Definition 4.9.

For each vertex of $\Gamma_{a}$ we define elements of
$\operatorname{Br}(\mathcal{D})$ inductively as follows:

  1. (1)

$\rho_{0}\coloneqq\Phi_{S_{0}}\Phi_{S_{-1}}$;

  2. (2)

$\rho_{i,1}\coloneqq\Phi_{(t_{i}^{1})}\rho_{0}\Phi_{(t_{i}^{1})}\rho_{0}^{-1}$
for $i=1,2,3$;

  3. (3)

$\rho_{i,j}\coloneqq\Phi_{(t_{i}^{j})}\rho_{i,j-1}\Phi_{(t_{i}^{j})}\rho_{i,j-1}^{-1}$
for $i=1,2,3$, $j=2,...,r_{i}-1$;

The assignment $\Phi_{S}\mapsto w_{S}$ defines a surjective homomorphism

$q\colon\operatorname{Br}(\mathcal{D})\twoheadrightarrow W$

######  Lemma 4.10.

The homomorphism $q$ maps the elements $\rho_{0},\rho_{i,j}$ to the elements
$r_{v_{0}},r_{v_{(i,j)}}\in T<W$.

###### Proof.

It follows from the definitions and from the fact that $q$ is a homomorphism.
∎

###  4.3. A t-structure on $\mathcal{D}$

This section aims to define a heart of a bounded t-structure $\mathcal{A}$ on
$\mathcal{D}$. To do so, we need to recall the McKay correspondence.

######  Definition 4.11.

A $\mu_{r}$-equivariant quotient sheaf
$\mathcal{O}_{\operatorname{Tot}(\omega_{E})}\twoheadrightarrow F$ is a
$\mu_{r}$-_cluster_ if its $\mathbb{C}[\mu_{r}]$-module structure is
isomorphic to the regular representation of $\mu_{r}$. We regard $F$ as an
element of $\operatorname{Coh}(\operatorname{Tot}(\omega_{X}))$.

Let
$Y^{\prime}\coloneqq\mu_{r}$-$\operatorname{Hilb}(\operatorname{Tot}(\omega_{X}))$
be the scheme parameterizing $\mu_{r}$-clusters on
$\operatorname{Tot}(\omega_{X})$. Then, $Y^{\prime}$ is a crepant resolution
of $\operatorname{Tot}(\omega_{X})$ [9]. We denote by $X^{\prime}\coloneqq
X\cup(\cup_{i,j}C_{i,j})$ the union of the strict transform of $X$ and of the
exceptional loci, and by $C_{i}$ the union $\cup_{j}C_{i,j}$. The curve
$X^{\prime}$ has a component isomorphic to $X$ and chains of rational curves
$C_{i,j=1,...,r_{i}-1}$ attached to $X$ at the point $p_{i}$.

There is an equivalence

$\Psi\colon D(Y^{\prime})\to D(\operatorname{Tot}(\omega_{X}))$

which in turn induces an equivalence between $\mathcal{D}$ and the full
triangulated subcategory $\mathcal{D}^{\prime}$ of sheaves supported on
$X^{\prime}$. Under the equivalence $\Psi$, we have

$\begin{split}\mathcal{O}_{C_{i,j}}(-1)\longmapsto t_{i}^{j};\\\
\mathcal{O}_{C_{i}}(C_{i})[1]\longmapsto t_{i}^{0};\\\
\mathcal{O}_{X^{\prime}}\longmapsto\mathcal{O}_{X}.\end{split}$

These conditions, together with the fact that $\Psi$ sends skyscraper sheaves
of $Y^{\prime}$ to clusters of $\operatorname{Tot}(\omega_{X})$, determines
$\Psi$ on $\mathcal{D}^{\prime}$.

Let
$\mathcal{B}^{\prime}=\operatorname{Coh}(Y^{\prime})\cap\mathcal{D}^{\prime}$
be the heart of the standard bounded t-structure in $\mathcal{D}^{\prime}$.
Then, define

$\mathcal{A}\coloneqq\Psi\mathcal{B}^{\prime}.$

Since $\Psi$ is an equivalence, the category $\mathcal{A}$ is the heart of a
bounded t-structure on $\mathcal{D}$.

######  Lemma 4.12.

The category $\mathcal{A}$ is Noetherian.

###### Proof.

This is straightforward, because $\mathcal{B}^{\prime}$ is Noetherian. ∎

######  Lemma 4.13.

Clusters in $\mathcal{A}$ are simple objects of class $a$.

###### Proof.

If $F$ is a cluster contained in $\mathcal{A}$, then $F=\Psi(\mathcal{O}_{t})$
for some $t\in X^{\prime}$, by definition of $\Psi$. Skyscraper sheaves are
simple in $\mathcal{B}^{\prime}$, and so is $F$ in $\mathcal{A}$. Since free
orbits are clusters, all clusters have class $a=[\mathbb{C}_{p}]$ in
$K(\mathcal{D})$. ∎

Before we proceed to a classification of objects in $\mathcal{A}$, we need an
alternative description of $\mathcal{A}$ as a tilt of the heart of the
standard t-structure on $\mathcal{D}$. Define $\mathcal{F}^{\prime}$ to be the
full subcategory of $\mathcal{B}^{\prime}$ generated by subsheaves of the
normal bundles $\mathcal{O}_{C_{i}}(C_{i})$ of the exceptional curves
$C_{i}\coloneqq\cup_{j=1}^{r_{i}-1}C_{i,j}$:

$\mathcal{F}^{\prime}=\langle
F\,|\,F\subset\mathcal{O}_{C_{i}}(C_{i})\in\mathcal{B}^{\prime}\text{ for
}i=1,2,3\rangle$

and $\mathcal{T}^{\prime}$ to be its left orthogonal in
$\mathcal{B}^{\prime}$. Denote by $\mathcal{F}$ (resp. $\mathcal{T}$) the
subcategories $\Psi\mathcal{F}^{\prime}$ (resp. $\Psi\mathcal{T}$) of
$\mathcal{A}$.

######  Proposition 4.14 (cfr. [31, Lemma 3.2]).

The pair of subcategories $(\mathcal{T}^{\prime},\mathcal{F}^{\prime})$ is a
torsion pair in $\mathcal{B}^{\prime}$. Therefore, the pair
$(\mathcal{T},\mathcal{F})$ is a torsion pair in $\mathcal{A}$ and
$\langle\mathcal{F}[1],\mathcal{T}\rangle=\mathcal{B}$.

###### Proof.

We need to show that every sheaf $E\in\mathcal{B}^{\prime}$ fits in a short
exact sequence

$T\to E\to F$

with $T\in\mathcal{T}$, $F\in\mathcal{F}$. If $E\in\mathcal{T}$, we are done.
Otherwise, $\operatorname{Hom}(E,\mathcal{F})\neq 0$, so there exists
$F_{1}\in\mathcal{F}$ fitting in a short exact sequence

$M_{1}\to E\to F_{1}.$

If $\operatorname{Hom}(M_{1},\mathcal{F})\neq 0$, repeat this process, and
obtain

$M_{2}\to M_{1}\to F_{2}.$

By iterating this, we get a chain of inclusions

$...\subset M_{k}\subset M_{k-1}\subset...\subset M_{1}\subset E$

with quotients in $\mathcal{F}$. Then, the chain must terminate by Lemma 4.15.
This means that there exists $n$ for which
$\operatorname{Hom}(M_{n},\mathcal{F})=0$. Let $F$ be the cokernel of the
inclusion $M_{n}\subset E$, then the sequence

$M_{n}\to E\to F$

is the desired one. The fact that $\Psi$ is an equivalence implies the
statement about $\mathcal{A}$. By construction, all objects in
$\langle\mathcal{F}[1],\mathcal{T}\rangle$ are sheaves, so we can apply Lemma
2.3 and conclude $\langle\mathcal{F}[1],\mathcal{T}\rangle=\mathcal{B}$. ∎

######  Lemma 4.15 (cfr. [31, Lemma 3.1]).

If there is a series of inclusions in $\mathcal{B}^{\prime}$, say

$...\subset M_{k}\subset M_{k-1}\subset...\subset M_{0}$

whose quotients lie in $\mathcal{F}$, then the sequence must eventually
stabilize.

###### Proof.

First, we may assume that all the quotients $F_{k}$ are supported on one curve
$C\coloneqq C_{i}$. Moreover:

######  Claim.

We may assume that for all $k$, the quotients $F_{k}$ are torsion free sheaves
$L_{k}\subset\mathcal{O}_{C}{(C)}$, such that $L_{k}$ has connected support
$D_{k}\subset C$.

Indeed, by definition every $F_{k}$ admits a surjection to some
$L_{k}\subset\mathcal{O}_{C}(C)$. By restricting $L_{k}$ to one of the
connected components $D_{k}$ of its support, we may assume that $L_{k}$ has
connected support. So we have quotients

$F_{k}\twoheadrightarrow L_{k}$

which define exact sequences

$0\to M_{k}^{(1)}\to M_{k}\to L_{k}\to 0.$

The quotient $F_{k}^{(1)}$ of $M_{k+1}\to M_{k}^{(1)}$ fits into an exact
sequence

$F_{k}^{(1)}\to F_{k}\to L_{k}$

where
$\operatorname{ch}_{{1}}{(F_{k}^{(1)})}=\operatorname{ch}_{{1}}{(F_{k})}-\operatorname{ch}_{{1}}{(L_{k})}$
is a positive linear combination $\sum a_{j}[C_{i,j}]$ with coefficients
strictly smaller than those of $\operatorname{ch}_{{1}}{(F_{k})}$. We can then
repeat this process for the map $M_{k+1}\to M_{k}^{(1)}$ until we get a finite
chain of inclusions

$M_{k+1}\subset M_{k}^{(n)}\subset...\subset M_{k}^{(1)}\subset M_{k}$

satisfying the statement of the claim.

We proceed to show that the sequence of inclusions must terminate with an
induction on the length $l$ of the chain of rational curves $C$.

In order to see this, apply the functor
$\operatorname{Hom}(-,\mathcal{O}_{C}(C))$ to the short exact sequence

(6) $0\to M_{k+1}\to M_{k}\to L_{k}\to 0.$

For $L_{k}=\mathcal{O}_{C}(C)$, one computes
$\operatorname{ext}^{1}(\mathcal{O}_{C}(C),\mathcal{O}_{C}(C))=0$, hence
$\operatorname{Hom}(M_{i},\mathcal{O}_{C}(C))>\operatorname{Hom}(M_{i+1},\mathcal{O}_{C}(C))$.

If $L_{k}\subsetneq\mathcal{O}_{C}(C)$, one has

(7)
$\operatorname{Ext}^{2}(L_{k},\mathcal{O}_{C}(C))\simeq\operatorname{Hom}(\mathcal{O}_{C}(C),L_{k})=0,$

and obtains

(8)
$\begin{split}\hom(M_{k},\mathcal{O}_{C}(C))-\hom(M_{k+1},\mathcal{O}_{C}(C))=\\\
\chi(L_{k},\mathcal{O}_{C}(C))+\left(\operatorname{ext}^{1}(M_{k},\mathcal{O}_{C}(C))-\operatorname{ext}^{1}(M_{k+1},\mathcal{O}_{C}(C))\right).\end{split}$

Observe that $\chi(L_{k},\mathcal{O}_{C}(C))=-(D_{k}).C\geq 0$ by Hirzebruch-
Riemann-Roch, and that

$\operatorname{ext}^{1}(M_{k},\mathcal{O}_{C}(C))-\operatorname{ext}^{1}(M_{k+1},\mathcal{O}_{C}(C))\geq
0$

because of (7).

If $l=1$, we must have $D_{k}=C$ and $-D_{k}.C=2$. This shows that if
$L_{k}\neq 0$, then
$\operatorname{Hom}(M_{k},\mathcal{O}_{C}(C))>\operatorname{Hom}(M_{k+1},\mathcal{O}_{C}(C))$,
whence the chain of subobjects must terminate.

If $l>1$, the only way the sequence does not terminate is that all $L_{k}$
satisfy $D_{k}.C=0$. This is only possible if no $D_{k}$ contains the terminal
curves of the chain, $C_{1}$ and $C_{l}$, in their support. In other words,
$L_{k}\subset\mathcal{O}_{C}(C)_{|C^{\prime}}\simeq\mathcal{O}_{C^{\prime}}(C^{\prime})$
where $C^{\prime}=\cup_{j=2}^{l-1}C_{j}$ is a shorter chain. Then, we can
repeat the argument above applying the functor
$\operatorname{Hom}(-,\mathcal{O}_{C^{\prime}}(C^{\prime}))$ to the sequences
(6). Eventually, the problem is reduced to the case $l=1$, and the process
must terminate. ∎

Next, we give a classification of objects in $\mathcal{A}$ for elliptic
orbifold quotients. Given a subchain of rational curves $D\subseteq C$, there
exists a maximal subsheaf $L_{D}\subseteq\mathcal{O}_{C}(C)$ supported on $D$.

######  Lemma 4.16.

Fix $C=C_{i}$, let $D\subseteq C$ be a subchain of rational curves, and let
$L_{D}$ as above. Write $C_{d_{1}},...,C_{d_{l}}$ for the irreducible
components of $D$ (with $(d_{1},...,d_{l})$ consecutive elements of
$\\{1,...,r_{i}-1\\}$). Then $L_{D}$ is obtained from
$\mathcal{O}_{C_{d_{1}}}(-2)$ with repeated extensions by the sheaves
$\mathcal{O}_{C_{d_{i}}}(-1)$, with $i=d_{2},...,d_{l}$. In particular, there
is a short exact sequence

(9) $L_{D}\to L^{\prime}_{D}\to\mathcal{O}_{t}$

where $t\in C_{d_{1}}$ and $L^{\prime}_{D}$ is obtained by repeated extensions
of $\mathcal{O}_{C_{d_{i}}}(-1)$, with $i=d_{1},...,d_{l}$.

###### Proof.

Be proceed by induction on the length $l$ of the chain $D$. If $l=1$ and
$D=C_{d}$, one readily verifies that $L_{D}\simeq\mathcal{O}_{C_{d}}(-2)$.
Suppose then that $l>1$. Then, observe that $L_{D}$ restricts to $C_{d_{l}}$
to a line bundle of degree $-1$, because either $d_{l}<r_{i}-1$, and then
sections of $L_{D}$ must vanish at the intersection $C_{d_{l}}\cap
C_{d_{l}+1}$ or because $d_{l}=r_{i}-1$, and $C$ has degree $-1$ on
$C_{r_{i}-1}$. The kernel of this restriction is exactly the maximal subsheaf
of $\mathcal{O}_{C}(C)$ supported on $\overline{D-C_{d_{l}}}$. In other words,
$L_{D}$ fits in a short exact sequence

$L_{\overline{D-C_{d_{l}}}}\to L_{D}\to\mathcal{O}_{C_{d_{l}}}(-1)$

so by induction $L_{D}$ has the asserted structure.

For the second statement, fix a point $t\in C_{d_{1}}$ away from the
intersections, and consider the cokernel

$(\epsilon)\colon\quad\mathcal{O}_{C_{d_{1}}}(-2)\to L_{D}\to R_{D}.$

From the sequence

$\mathcal{O}_{C_{d_{1}}}(-2))\to\mathcal{O}_{C_{d_{1}}}(-1))\to\mathcal{O}_{t}$

one sees that
$\operatorname{Ext}^{1}(R_{D},\mathcal{O}_{C_{d_{1}}}(-2))\simeq\operatorname{Ext}^{1}(R_{D},\mathcal{O}_{C_{d_{1}}}(-1))$
because $t\notin\text{Supp}R_{D}$. Pushing forward the extension class
$(\epsilon)$ to $\operatorname{Ext}^{1}(R_{D},\mathcal{O}_{C_{d_{1}}}(-1))$
produces an object $L_{D}^{\prime}$ as in the statement. ∎

######  Lemma 4.17.

Suppose an object $T\in\mathcal{A}$ is supported on an orbifold point $p_{i}$.
Then $T$ is obtained by repeated extensions of the following objects:

  1. (i)

$t_{i}^{j}$ with $j\neq 0$;

  2. (ii)

clusters supported at $p_{i}$;

  3. (iii)

$N[-1]$ where $N$ is a sheaf sitting in a sequence

$M\to N\to N^{\prime},$

where $M$ is obtained by repeated extensions of clusters (possibly $M=0$), and
$N^{\prime}$ is a proper quotient of a cluster.

###### Proof.

This is equivalent to classifying sheaves of $\mathcal{B}^{\prime}$ supported
on $C\coloneqq C_{i}$. First, we consider sheaves of $\mathcal{F}^{\prime}$. A
sheaf in $\mathcal{F}^{\prime}$ is an extension of subsheaves
$L\subset\mathcal{O}_{C}(C)$ with connected support. Any such inclusion must
factor thorugh an inclusion $L\subseteq L_{D}$, where $L_{D}$ is as in Lemma
4.16 and the cokernel $L_{D}/L$ is torsion. We have that $\Psi(L)[1]$ and
$\Psi(L_{D})[1]$ are sheaves on $X$, so applying the McKay functor to

$L\to L_{D}\to L_{D}/L$

we obtain a short exact sequence of sheaves in $\mathcal{B}$:

$M\to\Psi(L)[1]\to\Psi(L_{D})[1],$

where $M$ is obtained by repeated extensions of clusters. Now we claim that
$\Psi(L_{D})[1]$ is a proper quotient of a cluster. In fact, apply $\Psi$ to
the exact sequence (9) of Lemma 4.16: we have
$T\coloneqq\Psi(\mathcal{O}_{t})$ is a cluster, and $\Psi(L^{\prime}_{D})$ is
a sheaf obtained by repeated extensions of $t_{i}^{j}$, $j\neq 0$. This yields
a short exact sequence in $\mathcal{B}$

$0\to\Psi(L^{\prime}_{D})\to T\to\Psi(L_{D})[1]\to 0$

which exhibits $\Psi(L_{D})[1]$ as the quotient of a cluster. This exhausts
part (iii).

Now, consider a sheaf $B\in\mathcal{T}^{\prime}$. The torsion part $T(B)$ of
$B$ is obtained by repeated extensions of points, so $\Psi(T(B))$ is as in
part (ii). We may then assume that $B$ is torsion free with connected support.
If $B$ is supported on a single irreducible component $C_{i}$, then $B$ is a
sum of line bundles of the form $\mathcal{O}_{C_{i}}(k)$. Since
$\operatorname{Hom}(B,\mathcal{F}^{\prime})=0$, we must have $k>-2$. Then
$\Psi(B)$ is obtained as an extension of$t_{i}^{j}$ by clusters. If $B$ is
supported on more than one irreducible component, suppose that $C_{j}$ is a
terminal component of the support of $B$ and consider the restriction of $B$
to $C_{j}$. Then there is an exact sequence

$B^{\prime}\to B\to B_{|C_{j}}$

where $B^{\prime}$ is supported on a shorter chain. $B_{|C_{j}}$ is supported
on one irreducible curve, so it is as above. If
$B^{\prime}\in\mathcal{T}^{\prime}$, we repeat this procedure. Otherwise,
$B^{\prime}$ fits in a short exact sequence of sheaves

$B^{\prime\prime}\to B^{\prime}\to F$

with $B^{\prime\prime}\in\mathcal{T}^{\prime}$ and $F\in\mathcal{F}$. Since we
classified sheaves in $\mathcal{F}^{\prime}$ above, we can assume that
$B^{\prime}\in\mathcal{T}^{\prime}$, and conclude by induction on the length
of the supporting chain. ∎

As a consequence of the results in this section, we obtain the following
description of objects in $\mathcal{A}$:

######  Proposition 4.18.

Objects in $\mathcal{A}$ are obtained by repeated extensions from:

  1. (i)

line bundles on $X$;

  2. (ii)

skyscraper sheaves $\mathcal{O}_{q}$ for $q\in X-\cup\\{p_{i}\\}$;

  3. (iii)

torsion sheaves supported on orbifold points, classified in Lemma 4.17.

##  5\. Stability conditions on $\mathcal{D}$

###  5.1. The fundamental region $U$

Recall the notation introduced in Section 3 and the identification
$K(\mathcal{D})_{\mathbb{R}}\simeq F$. Then consider the central charge map

$\pi\colon\operatorname{Stab}(\mathcal{D})\to\operatorname{Hom}(F,\mathbb{C}).$

In this section, we construct stability conditions and investigate the image
of $\operatorname{Stab}(\mathcal{D})$ under the map $\pi$.

######  Proposition 5.1.

For every point $Z$ in the fundamental domain $D\subset\mathbb{E}$ there
exists a unique stability condition $(Z,\mathcal{A})$. These stability
conditions form a region $U\subset\operatorname{Stab}(\mathcal{D})$ which maps
homeomorphically to $D$ under the central charge map.

###### Proof.

Pick $Z\in D_{\tau}\subset D\subset\mathbb{E}$. The class of every object in
$\mathcal{A}$ is a positive linear combination of classes of objects listed in
Prop. 4.18. Then, the definition of $D_{\tau}$ shows that
$Z(\mathcal{A})\subset\overline{\mathbb{H}}$, in other words, $Z$ is a
stability function on $\mathcal{A}$. Since $\mathcal{A}$ is Noetherian (Lemma
4.12), and the image of $\operatorname{Im}Z$ is discrete by construction, then
$Z$ has the Harder-Narasimhan property by Prop. 2.5.

Again by Prop. 4.18, we see that the image of $Z$ is discrete, so the support
property is automatically satisfied. ∎

######  Lemma 5.2.

All $t_{i}^{j}$, $j\neq 0$, and all line bundles $\mathcal{O}_{X}(d)$ are
$\sigma$-stable for $\sigma\in U$.

###### Proof.

Let $S$ be one of the objects above. A short exact sequence

(10) $K\to S\to Q$

in $\mathcal{A}$ corresponds under the McKay functor to a short exact sequence
of sheaves on the crepant resolution

$K^{\prime}\to\Psi^{-1}S\to Q^{\prime}.$

On the other hand, $\Psi^{-1}S$ is either an object of the form
$\mathcal{O}_{C_{i,j}}(-1)$ or a line bundle on $X$. In either case, the only
quotients of $\Psi^{-1}S$ are obtained by repeated extensions of skyscraper
sheaves, so $Q\in\mathcal{A}$ is semistable of phase 1. Therefore $S$ is
$\sigma$-stable. ∎

###  5.2. Group actions and the image of the central charge map

In this section, we define a certain region
$\operatorname{Stab}^{\dagger}(\mathcal{D})$ of the stability manifold. We
define group actions which preserve
$\operatorname{Stab}^{\dagger}(\mathcal{D})$, and we study its image under the
central charge map.

######  Definition 5.3.

Let $\operatorname{Stab}^{\circ}(\mathcal{D})$ be the connected component of
$\operatorname{Stab}(\mathcal{D})$ containing $U$. We define
$\operatorname{Stab}^{\dagger}(\mathcal{D})$ as

(11)
$\operatorname{Stab}^{\dagger}(\mathcal{D})\coloneqq\left\\{\sigma=(Z,\mathcal{P})\in\operatorname{Stab}^{\circ}(\mathcal{D})\quad\middle|\quad(\ast)\colon\operatorname{Im}\frac{Z(b)}{Z(a)}>0\right\\}$

In fact, all stability conditions in
$\operatorname{Stab}^{\circ}(\mathcal{D})$ satisfy $(\ast)$, and we have
$\operatorname{Stab}^{\circ}(\mathcal{D})=\operatorname{Stab}^{\dagger}(\mathcal{D})$.
The proof of this fact uses our wall-crossing results, and is postponed to
Section 6.3. As a consequence we have:

######  Proposition 5.4.

The region $\operatorname{Stab}^{\dagger}(\mathcal{D})$ is a connected
component of $\operatorname{Stab}(\mathcal{D})$.

Then we have:

######  Lemma 5.5.

Let $\sigma=(\mathcal{A}^{\prime},Z)$ be a point in the boundary of $U$. Then
$\sigma$ lies in the union of lifts $\tilde{W}_{v,\pm}$,
$\tilde{Y}_{(i,j),\pm}$ of walls $W_{v,\pm}$, $Y_{(i,j),\pm}$.

###### Proof.

By the discussion in Sec. 3.4, the only other possibility is that
$\operatorname{Im}\frac{Z(b)}{Z(a)}=0$. But this is excluded by condition
$(\ast)$. ∎

Next, we consider two group actions on
$\operatorname{Stab}^{\dagger}(\mathcal{D})$, which lift the actions of
$\mathbb{C}^{*}$ and $W$ on $X_{reg}$.

There is a $\mathbb{C}$-action on $\operatorname{Stab}(\mathcal{D})$ defined
as follows. For $t\in\mathbb{C}$ and
$\sigma=(Z,\mathcal{P})\in\operatorname{Stab}(\mathcal{D})$, define
$t\cdot(Z,\mathcal{P})=(Z^{\prime},\mathcal{P}^{\prime})$, where

$Z^{\prime}(E)\coloneqq e^{-i\pi t}Z(E)\;\text{ and
}\;\mathcal{P}^{\prime}(\phi)\coloneqq\mathcal{P}(\phi+\operatorname{Re}t).$

The group $\operatorname{Aut}(\mathcal{D})$ of autoequivalences also acts on
$\operatorname{Stab}(\mathcal{D})$: for
$\Phi\in\operatorname{Aut}(\mathcal{D})$ and
$\sigma=(Z,\mathcal{P})\in\operatorname{Stab}(\mathcal{D})$, define
$\Phi\cdot(Z,\mathcal{P})=(Z^{\prime},\mathcal{P}^{\prime})$ as the stability
condition with

$Z^{\prime}(E)\coloneqq Z(\Phi^{-1}E)\;\text{ and
}\;\mathcal{P}^{\prime}(\phi)\coloneqq\Phi\mathcal{P}(\phi).$

The following discussion shows that the autoequivalences $\Phi_{S_{m}}$
preserve $\operatorname{Stab}^{\dagger}(\mathcal{D})$, so that
$\operatorname{Br}(\mathcal{D})$ acts on
$\operatorname{Stab}^{\dagger}(\mathcal{D})$.

######  Lemma 5.6.

Let $\sigma=(\mathcal{A}^{\prime},Z)$ be a point in the boundary of $U$
contained in a unique wall among the $\tilde{W}_{v,\pm}$’s. Then there is an
element $T\in\operatorname{Br}(\mathcal{D})$ such that $T\sigma$ also lies in
the boundary of $U$. More precisely, we may pick $T=\Phi_{S_{m}}$ if
$\sigma\in\tilde{W}_{v,+}$, and $T=\Phi^{-1}_{S_{m}}$ if
$\sigma\in\tilde{W}_{v,-}$.

###### Proof.

Suppose $\sigma\in\tilde{W}_{v,-}$. Set $S\coloneqq S_{m}$. Let $V$ be a small
neighborhood of $\sigma\in\operatorname{Stab}(\mathcal{D})$, and consider the
open subset

$V^{+}=\\{\tau=(\mathcal{B},Z)\in V\,|\,\operatorname{Im}Z(S)<0\\}.$

Arguing as in [8, Lemma 3.5], we claim that we can choose $V$ small enough so
that $\phi_{S}^{-1}(V^{+})\subset U$, hence $\Phi^{-1}_{S}\sigma$ lies in the
closure of $U$. Thus, we need to show that if $V$ is small enough, the heart
of any $\sigma^{\prime}=(\mathcal{A}^{\prime},Z^{\prime})\in V^{+}$ is equal
to $\Phi_{S}(\mathcal{A})\subset\mathcal{D}$. By Lemma 2.3, it is enough to
show that $\Phi_{S}(M)$ lies in the heart of any $\sigma^{\prime}\in V^{+}$,
for all the objects $M$ listed in Prop. 4.18.

We verify this on a case by case basis: assume first that $S=t_{i}^{j}$,
$j\neq 0$. Then:

Case 1. Suppose $L$ is a line bundle on $X$. There is a unique
$k\in\\{0,...,r_{i}\\}$ such that $\operatorname{Hom}(L,t_{i}^{k})\neq 0$.
Then $L$ is locally of the form $\mathcal{O}((k/r_{i})p_{i})$, and one
computes

$\operatorname{Hom}^{\bullet}(t_{i}^{j},L)=\begin{cases}\mathbb{C}[-1]\mbox{
if }k=j\\\ \mathbb{C}[-2]\mbox{ if }k+i=j\\\ 0\mbox{ otherwise.}\end{cases}$

If $\operatorname{Hom}^{1}(t_{i}^{j},L)\neq 0$, then there is a non-split
short exact sequence in $\mathcal{A}$

$L\to\Phi_{S}L\to t_{i}^{j}.$

It follows that $\Phi_{S}L$ lies in the heart of $\sigma$ and its semistable
factors have phases in $(0,1)$. Choosing $V$ small enough ensures that this is
the case for all $\sigma^{\prime}\in V^{+}$ too.

If $\operatorname{Hom}^{2}(t_{i}^{j},L)\neq 0$ then $\Phi_{S}L$ fits in a
triangle

$L\to\Phi_{S}L\to t_{i}^{j}[-1],$

which implies that $\Phi_{S}L$ lies in $\mathcal{A}^{\prime}$, because so do
$L$ and $t_{i}^{j}[-1]$.

If $\operatorname{Hom}^{\bullet}(t_{i}^{j},L)=0$ then $\Phi_{S}L=L$ and the
same argument applies.

Case 2. The same argument applies to
$\Phi_{t_{i}^{j}}(\mathcal{O}_{q})=\mathcal{O}_{q}$ for all $q\notin R$, and
to all sheaves supported away from $p_{i}$;

Case 3. The only possibilities for $\Phi_{S}t_{i}^{k}$, $k\neq j,0$ are that
$\operatorname{Hom}^{\bullet}(t_{i}^{j},t_{i}^{k})=0$ or
$\operatorname{Hom}^{1}(t_{i}^{j},t_{i}^{k})=\mathbb{C}$. Both are analogous
to the case of a line bundle above. Consider $\Phi_{S}(S)=S[-1]$. Since $S$ is
$\sigma$-stable of phase 1, we may assume that $S$ is $\sigma^{\prime}$-stable
with phase at most 2. Moreover, $S$ must have phase bigger than 1 in
$\sigma^{\prime}$, so $S[-1]$ lies in the heart of $\sigma^{\prime}$.
Similarly, one sees that $\Phi_{S}t_{i}^{0}[-1]\in\mathcal{A}^{\prime}$.

Case 4. If $M$ is a cluster supported at $p_{i}$, then $M$ has a non-split
composition series with factors the $t_{i}^{j}$ for $j=0,...,r_{i}-1$, where
$t_{i}^{0}$ is the last factor. Then, $\Phi_{S}M$ has a non-split composition
series with all factors in $\mathcal{A}^{\prime}$ but the last one in
$\mathcal{A}^{\prime}[1]$, and $Z^{\prime}(\Phi_{S}M)=-Z^{\prime}(a)=-1$, so
$\Phi_{S}(M)\in\mathcal{A}^{\prime}$.

Case 5. It remains to show the claim for $N[-1]$ where $N$ is the proper
quotient of a cluster $M$, with kernel $K$. Write the triangle

(12) $M[-1]\to N[-1]\to K$

and apply $\Phi_{S}$. By the discussion above,
$\Phi_{S}(K)\in\mathcal{A}^{\prime}$ since $K$ is obtained by repeated
extensions of $t_{i}^{j}$’s with $j>0$, and $\Phi_{S}(M)[-1]$ is stable of
phase 0. Then $\Phi_{S}(N)[-1]\in\mathcal{A}^{\prime}$, because the triangle
(12) does not split.

Similar computations show that $\Phi_{S}(M)\in\mathcal{A}^{\prime}$ for all
$M\in\mathcal{A}$ and $S=\mathcal{O}_{X}$. A symmetric argument settles the
case $\sigma\in\tilde{W}_{v,-}$. ∎

######  Lemma 5.7.

Let $\sigma=(\mathcal{A}^{\prime},Z)$ be a point in the boundary of $U$
contained in a unique wall among the $\tilde{Y}_{(i,j),\pm}$. Then there is an
element $T\in\operatorname{Br}(\mathcal{D})$ such that $T\sigma$ also lies in
the boundary of $U$. More precisely, we may pick $T=\rho_{j}$ if
$\sigma\in\tilde{Y}_{(i,j),+}$, and $T=\rho_{j}^{-1}$ if
$\sigma\in\tilde{Y}_{i,-}$.

###### Proof.

If $\sigma\in\tilde{Y}_{i,+}$, observe that we can choose a small neighborhood
$V$ of $\sigma$ in $\operatorname{Stab}(\mathcal{D})$ so that every $\tau\in
V$ has heart $\mathcal{A}$. Consider the open subset

$V^{\prime}=\\{\tau=(\mathcal{A},Z^{\prime})\in V\,|\,\tau\notin\bar{U}\\}$

For $\tau\in V^{\prime}$, we then have that
$\rho_{j}^{-1}Z^{\prime}=\rho_{j}^{-1}\operatorname{Re}Z^{\prime}+i\operatorname{Im}Z^{\prime}$
belongs to $D$. Then, it is enough to show $\rho_{j}(\mathcal{A})=\mathcal{A}$
to conclude $\rho_{j}\tau\in U$, so that $\rho_{j}\sigma$ lies in the closure
of $U$.

Using Prop. 4.18, one sees that $\mathcal{P}_{\sigma}(1)$ only contains
objects of class $a$ and its multiples. Since $\rho_{v}$ preserves the
imaginary part of $Z^{\prime}$ and fixes the class $a$, we have
$\mathcal{P}_{\tau}(1)=\mathcal{P}_{\sigma}(1)$. Then, the only possibility is
that for $v\in\lvert\Gamma_{a}\rvert$ one has
$\rho_{v}(\mathcal{A})=\mathcal{A}[2n]$, for some integer $n$. We prove that
$n$ must be 0. One readily checks

$\rho_{0}(\mathcal{O}_{X}(1))=\Phi_{\mathcal{O}_{X}}\Phi_{\mathcal{O}_{X}(1)}(\mathcal{O}_{X}(1))\simeq\Phi_{\mathcal{O}_{X}}(\mathcal{O}_{X}(1)[-1])=\mathcal{O}_{X}(-1).$

using Lemma 4.4. This implies that $\rho_{0}(\mathcal{A})=\mathcal{A}$. Now
one has

(13)
$\begin{split}\rho_{(i,1)}(\mathcal{O}_{X}(-1))&=\Phi_{(t_{i}^{1})}\rho_{0}\Phi_{(t_{i}^{1})}\rho_{0}^{-1}(\mathcal{O}_{X}(-1))\\\
&\simeq\Phi_{(t_{i}^{1})}\rho_{0}\Phi_{(t_{i}^{1})}(\mathcal{O}_{X}(1))\\\
&\simeq\Phi_{(t_{i}^{1})}\Phi_{\mathcal{O}_{X}}\Phi_{\mathcal{O}_{X}(1)}\Phi_{(t_{i}^{1})}(\mathcal{O}_{X}(1))\\\
&\simeq\Phi_{(t_{i}^{1})}\Phi_{\mathcal{O}_{X}}(t_{i}^{1})\\\
&\simeq(\mathcal{O}_{X}),\end{split}$

by repeatedly applying Lemma 4.4. For $\rho_{(i,j)}$, $j>1$, we claim
$\rho_{(i,j)}(\mathcal{O}_{X})\simeq\mathcal{O}_{X}$. This is a consequence of
the fact that $\mathcal{O}_{X}(d)$ is orthogonal to $t_{i}^{j}$ for $d=0,-1$,
all $i$ and all $j>1$. Indeed, one computes

(14)
$\begin{split}\rho_{(i,2)}(\mathcal{O}_{X})&=\Phi_{(t_{i}^{2})}\rho_{(i,1)}\Phi_{(t_{i}^{2})}\rho_{(i,1)}^{-1}(\mathcal{O}_{X})\\\
&\simeq\Phi_{(t_{i}^{2})}\rho_{(i,1)}\Phi_{(t_{i}^{2})}(\mathcal{O}_{X}(-1))\\\
&\simeq\Phi_{(t_{i}^{2})}\rho_{(i,1)}(\mathcal{O}_{X}(-1))\\\
&\simeq\Phi_{(t_{i}^{1})}(\mathcal{O}_{X})\\\
&\simeq(\mathcal{O}_{X}),\end{split}$

and proves the same claim for $j>2$ inductively. This concludes the proof in
the case $\sigma\in\tilde{Y}_{i,+}$. The case $\sigma\in\tilde{Y}_{i,-}$ is
similar. ∎

Let $\pi$ be the restriction of the central charge map to
$\operatorname{Stab}^{\dagger}(\mathcal{D})$, and define
$\operatorname{Stab}^{\dagger}(\mathcal{D})^{N}$ to be

$\operatorname{Stab}^{\dagger}(\mathcal{D})^{N}\coloneqq\pi^{-1}\mathbb{E}$

######  Proposition 5.8.

For any $\sigma\in\operatorname{Stab}^{\dagger}(\mathcal{D})^{N}$, there is an
autoequivalence $\Phi\in\operatorname{Br}(\mathcal{D})$ such that
$\Phi\sigma\in U$.

###### Proof.

Same as the proof of Prop. 4.13 in [15]. ∎

Let $\pi^{-1}(X_{reg})^{\dagger}$ be the connected component of
$\pi^{-1}X_{reg}$ which contains $U$. Then

######  Corollary 5.9.

For any $\sigma\in\pi^{-1}(X_{reg})^{\dagger}$, there is an autoequivalence
$\Phi\in\operatorname{Br}(\mathcal{D})$ and $k\in\mathbb{C}$ such that
$(k\cdot\Phi)(\sigma)\in U$.

###### Proof.

See [15, Cor. 4.14]. ∎

######  Lemma 5.10.

The image of
$\pi\colon\operatorname{Stab}^{\dagger}(\mathcal{D})\to\operatorname{Hom}(F,\mathbb{C})$
contains $X_{reg}$.

###### Proof.

The component $\operatorname{Stab}^{\dagger}(\mathcal{D})$ contains the orbit
under $\mathbb{C}$ and $\operatorname{Br}(\mathcal{D})$ of $U$. Since the
actions of $\mathbb{C}$ and $\operatorname{Br}(\mathcal{D})$ lift those of
$\mathbb{C}^{*}$ and $W$ on $\operatorname{Hom}(F,\mathbb{C})$, the orbit of
$U$ under the actions of $\mathbb{C}$ and $\operatorname{Br}(\mathcal{D})$ is
mapped to $X_{reg}\subset\operatorname{Hom}(F,\mathbb{C})$. ∎

The next goal of our discussion is to prove the following:

######  Proposition 5.11.

The projection $\pi$ maps $\operatorname{Stab}^{\dagger}(\mathcal{D})$ onto
$X_{reg}$.

###### Proof.

By Lemma 5.10, it is enough to show that
$\pi(\operatorname{Stab}^{\dagger}(\mathcal{D}))\subseteq X_{reg}$, or
equivalently that
$\operatorname{Stab}^{\dagger}(\mathcal{D})\subseteq\pi^{-1}(X_{reg})^{\dagger}$.
To show this, it is enough to check that
$\operatorname{Stab}^{\dagger}(\mathcal{D})$ contains no boundary points of
$\pi^{-1}(X_{reg})^{\dagger}$. Any such boundary point
$\sigma=(Z,\mathcal{P})$ is projected to $Z\in\partial X_{reg}$. From the
definition of $X_{reg}$ in Section 3.3, there is a ray in $\mathbb{R}_{>0}(R)$
such that $Z(S)=0$, or $\operatorname{Im}\frac{Z(b)}{Z(a)}=0$.

In the latter case, condition $(*)$ ensures that
$\sigma\notin\operatorname{Stab}^{\dagger}(\mathcal{D})$. Suppose $\alpha$ is
a positive root such that $Z(\alpha)=0$. If
$\sigma\in\overline{\operatorname{Stab}^{\dagger}(\mathcal{D})}$, by
proposition 5.8 there is an element $\Phi\in\operatorname{Br}(\mathcal{D})$,
such that $\Phi\cdot\sigma=(Z^{\prime},\mathcal{P}^{\prime})\in\overline{U}$,
and $[\Phi]\alpha=\beta\in\Pi$. Then we have $Z^{\prime}(\beta)=0$. However,
by Lemma 5.2, for all $\beta\in\Pi$ there are objects of class $\beta$ which
are semistable for all stability conditions in $U$, hence $\Phi\cdot\sigma$
violates the support property, and therefore
$\sigma\notin\operatorname{Stab}^{\dagger}(\mathcal{D})$. ∎

######  Proposition 5.12.

The action of $\mathbb{Z}[2]\times\operatorname{Br}(\mathcal{D})$ on
$\operatorname{Stab}^{\dagger}(\mathcal{D})$ is free and properly
discontinuous.

###### Proof.

This is clear for the action of $\mathbb{Z}[2]$, so it is enough to check it
for $\operatorname{Br}(\mathcal{D})$.

First, we check that the action of $\operatorname{Br}(\mathcal{D})$ is free.
By Cor. 5.9, it is enough to show this for $\sigma\in U$. Assume then that
$\sigma=\Phi\sigma$ for some $\Phi\in\operatorname{Br}(\mathcal{D})$ and
$\sigma\in U$. At the level of K-theory, we have $[\Phi]^{-1}\cdot Z=Z$, hence
$[\Phi]=\operatorname{id}$. So $[\Phi(S_{m})]=[S_{m}]$ for all $m$. Up to
isomorphism, $S_{m}$ is the only object in $\mathcal{A}$ in its class (this is
readily observed translating $\mathcal{A}$ to $\Psi^{-1}\mathcal{A}$), hence
$\Phi(S_{m})\simeq S_{m}$ for all $m$. Then $\Phi=\operatorname{id}$ in
$\operatorname{Br}(\mathcal{D})$ by Lemma 5.13.

To show that the action of $\operatorname{Br}(\mathcal{D})$ is properly
discontinuous, it is enough to exhibit, for every non-trivial
$\Phi\in\operatorname{Br}(\mathcal{D})$ and every $\sigma\in U$, a
neighborhood $V$ of $\sigma$ such that $\Phi(V)\cap V=$. If
$[\Phi]\neq\operatorname{id}$, the existence of $V$ follows from Prop. 3.13.
If $[\Phi]=\operatorname{id}$, then it is a consequence of Lemma 2.7. ∎

######  Lemma 5.13.

Suppose $\Phi\in\operatorname{Br}(\mathcal{D})$ satisfies $\Phi(S)\simeq S$
for all $S\in\Pi$. Then $\Phi\simeq\operatorname{id}$.

###### Proof.

We consider $\Phi$ as an element of
$\operatorname{Aut}(D^{b}(\operatorname{Tot}(\omega_{X})))$, and we study the
equivalent problem of showing that

$\Phi^{\prime}\coloneqq\Psi^{-1}\circ\Phi\circ\Psi$

is the identity on $\operatorname{Aut}(D^{b}(Y^{\prime}))$, where $Y^{\prime}$
denotes the crepant resolution of $\operatorname{Tot}(\omega_{X})$, under the
assumption that elements of $\Psi^{-1}\Pi$ are fixed (recall the notation of
Section 4).

First, observe that for $p\in Y^{\prime}\setminus X^{\prime}$ we have
$\Phi(\mathcal{O}_{p})\simeq\mathcal{O}_{p}$ because all $S\in\Pi$ are
supported on $X$ and hence orthogonal to $\mathcal{O}_{p}$. If $p\in X\subset
X^{\prime}$, applying $\Phi$ to the short exact sequence

$0\to
i_{*}\mathcal{O}_{X}(-1)\xrightarrow{f}i_{*}\mathcal{O}_{X}\to\mathcal{O}_{p}\to
0$

one obtains a non zero map $\Phi(f)$ of pure one-dimensional sheaves, fitting
in a triangle

$i_{*}\mathcal{O}_{X}(-1)\xrightarrow{\Phi(f)}i_{*}\mathcal{O}_{X}\to\Phi(\mathcal{O}_{p}).$

This implies that $H^{-1}\Phi(\mathcal{O}_{p})=0$ and $\Phi(\mathcal{O}_{p})$
is a skyscraper supported at a point of $X$.

Now let $\\{p\\}=X\cap C_{i,1}$. Then the skyscraper supported at $p$ must be
fixed by $\Phi^{\prime}$, because it admits a restriction map
$\mathcal{O}_{C_{i,1}}(-1)\to\mathcal{O}_{p}$ and $\Phi^{\prime}$ fixes
$\mathcal{O}_{C_{i,1}}(-1)=\Psi^{-1}t_{i}^{1}$. Let $M_{p}$ denote the cluster
corresponding to $p$. Then $\Phi$ fixes $M_{p}$ because $\Phi^{\prime}$ fixes
$\mathcal{O}_{p}$. Moreover, $M_{p}$ has a unique composition series by the
$t_{i}^{j}$, which are all fixed by $\Phi$ except possibly $t_{i}^{0}$. Then
$\Phi$ must also fix $t_{i}^{0}$ for $i=1,2,3$.

Then, since every cluster has a composition series with factors the simple
sheaves $t_{i}^{j}$ and $\Phi$ fixes the $t_{i}^{j}$ for all
$j=0,...,r_{i}-1$, it must also send any cluster to a cluster. In other words,
$\Phi^{\prime}$ sends skyscraper sheaves of points on any exceptional curve
$C_{i}$ to skyscraper sheaves.

Once can then apply [Huy06, Cor. 5.23], which implies that there exists an
automorphism $\phi$ of $Y^{\prime}$ such that
$\Phi^{\prime}(\mathcal{O}_{t})\simeq\mathcal{O}_{\phi(t)}$ and
$\Phi^{\prime}\simeq(-\otimes\mathcal{L})\circ\phi_{*}$ for some line bundle
$\mathcal{L}$ on $Y^{\prime}$. The automoprhism $\phi$ is the identity,
because it is the identity on the dense open complement of $X^{\prime}$. The
Picard group of $\operatorname{Tot}(\omega_{X})$ is isomorphic to
$\operatorname{Pic\,}(X)\bigoplus(\oplus\mathbb{Z}\\{C_{i,j}\\})$ hence the
only line bundle fixing the $\Psi^{-1}(S)$ with $S\in\Pi$ is the trivial one.
Then, $\Phi^{\prime}\simeq\operatorname{id}$ as we wished to prove. ∎

###  5.3. Proof of main results

Denote by $\bar{\pi}$ the composition of the maps
$\operatorname{Stab}^{\dagger}(\mathcal{D})\xrightarrow{\pi}X_{reg}\to
X_{reg}/\tilde{W}$. Then we have:

######  Theorem 5.14.

The map

$\bar{\pi}\colon\operatorname{Stab}^{\dagger}(\mathcal{D})\to
X_{reg}/\tilde{W}$

is a covering map, and the group
$\mathbb{Z}[2]\times\operatorname{Br}(\mathcal{D})$ acts as group of deck
transformations.

###### Proof.

We only need to show that the quotient of
$\operatorname{Stab}^{\dagger}(\mathcal{D})$ by
$\mathbb{Z}[2]\times\operatorname{Br}(\mathcal{D})$ coincides with
$X_{reg}/\tilde{W}$. Equivalently, for every pair of stability conditions
$\sigma_{1}$, $\sigma_{2}$ satisfying
$\bar{\pi}(\sigma_{1})=\bar{\pi}(\sigma_{2})$, we need to exhibit elements
$[2n]\in\mathbb{Z}[2]$ and $\Phi\in\operatorname{Br}(\mathcal{D})$ such that
$\sigma_{1}=([2n]\cdot\Phi)(\sigma_{2})$.

By Corollary 5.9, it is enough to show this when $\sigma_{1}\in U$. Moreover,
there are elements $\Phi\in\operatorname{Br}(\mathcal{D})$, $k\in\mathbb{C}$,
such that $\sigma_{2}^{\prime}\coloneqq(k\cdot\Phi)(\sigma_{2})$ lies in $U$.
Then we have

$\pi(\sigma_{2}^{\prime})=[\Phi]\cdot e^{-i\pi
k}\cdot\pi(\sigma_{2})=[\Phi]\cdot e^{-i\pi k}\cdot\pi(\sigma_{1})$

in $D$. Since $U$ and $D$ are homeomorphic, this implies that
$[\Phi]=\operatorname{id}$, $k\in 2\mathbb{Z}$, and
$\sigma_{2}^{\prime}=\sigma_{1}$. ∎

Let
$\operatorname{Aut}^{\dagger}(\mathcal{D})\subset\operatorname{Aut}(\mathcal{D})$
be the subgroup of autoequivalence preserving the region
$\operatorname{Stab}^{\dagger}(\mathcal{D})$. Write
$\operatorname{Aut}^{\dagger}_{*}(\mathcal{D})$ for the quotient of
$\operatorname{Aut}^{\dagger}(\mathcal{D})$ by the subgroup of
autoequivalences which act trivially on
$\operatorname{Stab}^{\dagger}(\mathcal{D})$.

######  Corollary 5.15.

There is an isomorphism

$\operatorname{Aut}^{\dagger}_{*}(\mathcal{D})\simeq\mathbb{Z}[1]\times\left(\operatorname{Br}(\mathcal{D})\rtimes\operatorname{Aut}(\Gamma)\right),$

Where $Aut(\Gamma)$ acts on $\operatorname{Br}(\mathcal{D})$ by permuting the
generators.

###### Proof.

The proof is identical to that of [8, Cor. 1.5]. ∎

##  6\. Wall-crossing

This section is dedicated to the study of wall-crossing for objects in
$\mathcal{D}$. First, we produce stable objects for a certain stability
condition in $\operatorname{Stab}^{\dagger}(\mathcal{D})$. We then analyze
wall crossing for spherical and radical classes. We apply these results to
obtain a proof of Proposition 5.4. We keep the notation as above.

###  6.1. Stability conditions on $\operatorname{Coh}(X)$ and $\mathcal{B}$

Geigle and Lenzing define slope stability on a weighted projective line in
[11, Sec. 5]. Define a stability condition
$\tau_{0}^{\prime}\coloneqq(Z_{0},\operatorname{Coh}(X))\in\operatorname{Stab}(X)$
with

$Z_{0}=-\deg+i\operatorname{rk},$

where $\deg(\mathcal{O}_{p_{i}}\chi^{j})$ is defined to be $-\frac{1}{r_{i}}$
for all orbifold points $p_{i}$ and all $j=0,...,r_{i}-1$. Then, slope
stability is equivalent to $\tau_{0}^{\prime}$-stability on $X$. We say that a
root $\alpha\in R\cup\Delta_{im}$ is _positive_ if
$Z_{0}(\alpha)\in\mathbb{H}\cup\mathbb{R}_{<0}$. Results about
$\tau_{0}^{\prime}$-stability are summarized in [20]:

######  Theorem 6.1 ([20, Theor. 4.6]).

Let $X$ be as above, $\alpha\in R\cup\Delta_{im}$. Then:

  1. (i)

there exists an indecomposable sheaf $F$ of class $\alpha$ if and only if
$\alpha$ is a positive root;

  2. (ii)

the sheaf $F$ is unique up to isomorphism if $\alpha$ is a real root, and
varies in a one-parameter family if $\alpha$ is imaginary;

  3. (iii)

an indecomposable sheaf is $\tau_{0}^{\prime}$-semistable, and it is
$\tau_{0}^{\prime}$-stable if and only if $\alpha$ is primitive.

In virtue of Lemma 4.3, we can regard $Z_{0}$ as a map defined on
$K(\mathcal{D})$, and define a stability condition
$\tau_{0}\in\operatorname{Stab}(\mathcal{D})$ as $(Z_{0},\mathcal{B})$.
Observe that, by construction, $\tau_{0}$ lies in the boundary of a
fundamental chamber in $\operatorname{Stab}^{\dagger}(\mathcal{D})$. We say
that an object $E\in\mathcal{D}$ is _semi-rigid_ if
$\operatorname{ext}^{1}(E,E)=2$. Then we have:

######  Proposition 6.2.

Let $\alpha\in R\cup\Delta_{im}$ be a positive root. If $\alpha$ is a real
root, there exist a $\tau_{0}$-semistable spherical sheaf in $\mathcal{B}$ of
class $\alpha$. If $\alpha$ is imaginary, there is a one-parameter family of
semi-rigid $\tau_{0}$-semistable sheaves in $\mathcal{B}$ of class $\alpha$.
If $\alpha$ is primitive, we can replace semistability with stability.

###### Proof.

By Theorem 6.1, there exists a $\tau_{0}^{\prime}$-semistable sheaf
$E^{\prime}$ on $X$ of class $\alpha$. Let $E\coloneqq\iota_{*}(E^{\prime})$
be the indecomposable sheaf in $\mathcal{B}$ obtained by pushing forward
$E^{\prime}$. The sheaf $E$ is $\tau_{0}$-semistable: since $E$ is supported
on $X$ then so must be every subsheaf $S\subset E$. This implies that
$S=\iota_{*}S^{\prime}$ for some $S^{\prime}\in\operatorname{Coh}(X)$. Then,
$S$ destabilizes $E$ if and only if $S^{\prime}$ destabilizes $E^{\prime}$.

Next, we show that $E$ is spherical if $\alpha$ is a real root. As a
consequence of Theorem 6.1 we have that
$\operatorname{Ext}^{1}_{X}(E^{\prime},E^{\prime})=0$, hence
$\operatorname{Ext}^{1}_{\mathcal{B}}(E,E)=0$ by Lemma 4.2. On the other hand,
since $\alpha$ is real one must have $\chi(\alpha,\alpha)=2$, so $E$ is
spherical. Similarly, one argues that $E$ is semi-rigid if $\alpha$ is
imaginary. The claim about stability follows again from Theorem 6.1. ∎

###  6.2. Wall-crossing in $\operatorname{Stab}(\mathcal{D})$

The lattice $K(\mathcal{D})$ can be equipped with the Mukai pairing

$(\mathbf{v},\mathbf{w})\coloneqq-\chi(\mathbf{v},\mathbf{w}).$

The pairing has a rank 2 radical $\operatorname{rad}\chi$ generated by $a$ and
$b$, and it induces a negative definite pairing on
$K(\mathcal{D})/\operatorname{rad}\chi$, since the Euler form on
$K(\mathcal{D})/\operatorname{rad}\chi$ coincides with the Cartan matrix of
the root system $R_{f}$, which is positive definite.

Since $K(\mathcal{D})$ is negative semidefinite, the class $\mathbf{v}$ of a
stable object can only satisfy $\mathbf{v}^{2}=0$ or $\mathbf{v}^{2}=-2$. In
the first case, $\mathbf{v}$ belongs to $\operatorname{rad}\chi$, and we call
it a radical class. Classes with $\mathbf{v}^{2}=-2$ are called spherical
classes.

First, notice that since $K(\mathcal{D})$ is a discrete lattice, we have a
finiteness result for walls:

######  Proposition 6.3 ([1, Prop. 3.3]).

Let $\mathcal{D}$ be a triangulated category such that $K(\mathcal{D})$ is a
lattice of finite rank. Let
$\operatorname{Stab}^{*}(\mathcal{D})\subset\operatorname{Stab}(\mathcal{D})$
be a connected component of its space of stability conditions. Fix a primitive
class $\mathbf{v}\in K(\mathcal{D})$, and an arbitrary set $S\subset D$ of
objects of class $\mathbf{v}$. Then there exists a collection of walls
$W^{S}_{\mathbf{w}}$, with $\mathbf{w}\in K(\mathcal{D})$, with the following
properties:

  1. (a.)

Every wall $W^{S}_{\mathbf{w}}$ is a closed submanifold with boundary of real
codimension one;

  2. (b.)

The collection $W^{S}_{\mathbf{w}}$ is locally finite (i.e., every compact
subset $K\subset\operatorname{Stab}^{*}(\mathcal{D})$ intersects only a finite
number of walls);

  3. (c.)

For every stability conditions $(Z,\mathcal{P})\in W^{S}_{\mathbf{w}}$, there
exists a phase $\phi$ and an inclusion $F_{\mathbf{w}}\to E_{\mathbf{v}}$ in
$\mathcal{P}(\phi)$ with $[F_{\mathbf{w}}]=\mathbf{w}$ and some
$E_{\mathbf{v}}\in S$;

  4. (d.)

If $\mathcal{C}\subset\operatorname{Stab}^{*}(\mathcal{D})$ is a connected
component of the complement of $\cup_{\mathbf{w}\in
K(\mathcal{D})}W^{S}_{\mathbf{w}}$, and $\sigma_{1},\sigma_{2}\in\mathcal{C}$,
then an object $E_{\mathbf{v}}\in S$ is $\sigma_{1}$-stable if and only if it
is $\sigma_{2}$-stable.

Recall that $\sigma\in\operatorname{Stab}(\mathcal{D})$ is said to be
_generic_ with respect to $\mathbf{v}\in K(\mathcal{D})$ if $\sigma$ does not
lie on any of the walls of the wall-and-chamber decomposition associated to
$\mathbf{v}$. The goal of this section is to prove the following Theorem.

######  Theorem 6.4.

Let $\alpha\in R\subset K(\mathcal{D})$ be a positive root. Let
$\sigma\in\operatorname{Stab}^{\circ}(\mathcal{D})$ be generic with respect to
$\alpha$. Then, there exists a $\sigma$-stable object $E$ of class $\alpha$.
The object $E$ is rigid if $\alpha$ is a real root, and it varies in a family
if $\alpha$ is imaginary.

We will make use of the following well-known property of K3-categories.

######  Lemma 6.5 ([14, Prop. 2.9]).

Let $\sigma\in\operatorname{Stab}(\mathcal{D})$.

  1. (i)

If $E\in\mathcal{D}$ is spherical, then all of its $\sigma$-stable factors are
spherical;

  2. (ii)

if $E\in\mathcal{D}$ is semi-rigid, then all of its $\sigma$-stable factors
are spherical, except for possibly one semi-rigid factor.

Before moving forward, we recall a construction from [2]. Fix a primitive
class $\mathbf{v}\in K(\mathcal{D})$, let $S$ be the set of objects of
$\mathcal{D}$ of class $\mathbf{v}$, and let $W=W^{S}_{\mathbf{w}}$ be a wall
of the wall-and-chamber decomposition of $\operatorname{Stab}(\mathcal{D})$
associated to $\mathbf{v}$. Then we can associate to $W$ the rank 2 lattice
$H_{W}\subset K(\mathcal{D})$:

(15) $H_{W}=\left\\{\mathbf{w}\in
K(\mathcal{D})\mid\operatorname{Im}\frac{Z(\mathbf{v})}{Z(\mathbf{w})}=0\mbox{
for all }\sigma=(Z,\mathcal{P})\in W\right\\}.$

The rank of $H_{W}$ is at least 2 because it contains at least $\mathbf{v}$
and the linearly independent class $\mathbf{w}$ destabilizing at $W$. If it
had rank bigger than 2, the definition (15) would imply that $W$ has
codimension higher than 1.

For any $\sigma=(Z,\mathcal{P})\in W$, let $C_{\sigma}\subset
H_{W}\otimes\mathbb{R}$ be the cone spanned by classes $\gamma$ satisfying

$\mathbf{c}^{2}\geq-2\quad\mbox{and}\quad\operatorname{Im}\frac{Z(\mathbf{c})}{Z(\mathbf{v})}>0.$

We will refer to $C_{\sigma}$ as to the _cone of $\sigma$-effective classes_
in $H_{W}$.

####  6.2.1. Wall-crossing for spherical classes

######  Lemma 6.6.

Let $\mathbf{v}$ be a primitive spherical class in $K(\mathcal{D})$, and $W$
be a wall for $\mathbf{v}$. Then $H_{W}$ is a primitive lattice of rank two
generated by $\mathbf{v}$ and a spherical class $\mathbf{w}$. It is negative
definite (with respect to the restriction of the Mukai pairing). Moreover,
there are only three possibilities for the intersection form, and:

  1. (i)

if $(\mathbf{v},\mathbf{w})=0$, then $H_{W}$ contains no spherical classes
except for $\pm\mathbf{v}$ and $\pm\mathbf{w}$;

  2. (ii)

if $(\mathbf{v},\mathbf{w})=-1$, the only spherical classes in $H_{W}$ are
$\pm\mathbf{v}$, $\pm\mathbf{w}$, and $\pm(\mathbf{v}-\mathbf{w})$;

  3. (iii)

if $(\mathbf{v},\mathbf{w})=1$, the only spherical classes in $H_{W}$ are
$\pm\mathbf{v}$, $\pm\mathbf{w}$, and $\pm(\mathbf{v}+\mathbf{w})$.

###### Proof.

We have that $\mathbf{v}\in H_{W}$ has $\mathbf{v}^{2}<0$ and $\mathbf{w}$
must be a spherical class by Lemma. 6.5. So both $\mathbf{v}$ and $\mathbf{w}$
project to non-zero vectors in $K(\mathcal{D})/\operatorname{rad}$. The
intersection matrix of $H_{W}$ can be computed on
$K(\mathcal{D})/\operatorname{rad}$, where the Mukai pairing coincides with
the opposite of the Cartan intersection matrix, so it is negative definite.

The signature of the form implies that the determinant of the intersection
form be positive, which rules out all values of $(\mathbf{v},\mathbf{w})$
except for $0$ and $\pm 1$. The spherical classes are the integer solutions of

$-2=(x\mathbf{v}+y\mathbf{w})^{2}=-2x^{2}-2y^{2}+2(\mathbf{v},\mathbf{w})xy$

in these three cases. ∎

Let $W$ be a potential wall for $\mathbf{v}$. Then, we denote by $\sigma_{0}$
a stability condition which only lies on the wall $W$, and consider a path in
$\operatorname{Stab}(\mathcal{D})$ passing through $\sigma_{0}$ and connecting
$\sigma^{+}$ and $\sigma_{-}$, two stability conditions lying in adjacent
chambers.

######  Lemma 6.7.

For $W$ as above, suppose that there exists an indecomposable
$\sigma_{0}$-semistable spherical object $E$ of class $\mathbf{v}$. Then there
is a $\sigma^{+}$-stable spherical object $E^{+}$ of class $\mathbf{v}$.
Likewise, there exist a $\sigma^{-}$-stable object $E^{-}$ of class
$\mathbf{v}$.

###### Proof.

By Lemma 6.5, the Jordan-Hölder factors of $E$ are spherical objects. In other
words, $\mathbf{v}$ can be written as a sum of spherical classes in
$C_{\sigma_{0}}$. If $E$ is $\sigma_{0}$-stable, there is nothing to prove.
Otherwise, Lemma 6.6 shows that, up to the sign of $\mathbf{w}$, $E$ has a
Jordan-Hölder filtration

$B\to E\to A$

where $B$, $A$ have class $\mathbf{w}$ and $\mathbf{v}-\mathbf{w}$,
respectively. Observe that
$\operatorname{Ext}^{1}(A,B)=\operatorname{Ext}^{1}(B,A)\neq 0$ since $E$ is
indecomposable, and denote by $E^{\prime}$ the non-trivial extension

$A\to E^{\prime}\to B.$

If $\phi_{\sigma^{+}}(\mathbf{v}-\mathbf{w})>\phi_{\sigma^{+}}(\mathbf{w})$
set $E^{+}=E$. If
$\phi_{\sigma^{+}}(\mathbf{v}-\mathbf{w})<\phi_{\sigma^{+}}(\mathbf{w})$, set
$E^{+}=E^{\prime}$. In any case, $E^{+}$ satisfies the assumptions of [2,
Lemma 9.3], and hence is $\sigma^{+}$-stable. ∎

####  6.2.2. Wall-crossing for radical classes

######  Lemma 6.8.

Let $\mathbf{v}$ be a primitive radical class in $K(\mathcal{D})$, and $W$ be
a potential wall for $\mathbf{v}$. Then the intersection matrix of $H_{W}$ is
either the zero matrix or

$\begin{pmatrix}0&0\\\ 0&-2\end{pmatrix}.$

If the intersection form is zero, $H$ is contained in
$\operatorname{rad}\chi$, it contains no spherical classes, and $W$ is not a
wall. Otherwise, $H$ contains a spherical class $\mathbf{w}$.

###### Proof.

Another generator of $H_{W}$, $\mathbf{w}$, is either semi-rigid or spherical
by Lemma 6.5. If it is semi-rigid, $H$ contains no spherical classes. Then
every $\sigma_{0}$-semistable object $E$ of class $\mathbf{v}$ must be stable
on $W$, because it can only have one Jordan-Hölder factor. ∎

######  Lemma 6.9.

For $W$ as above, suppose that there exists an indecomposable
$\sigma_{0}$-semistable semi-rigid object $E$ of class $\mathbf{v}$. Then
there is a $\sigma^{+}$-stable semi-rigid object $E^{+}$ of class
$\mathbf{v}$. Likewise, there exist a $\sigma^{-}$-stable semi-rigid object
$E^{-}$ of class $\mathbf{v}$.

###### Proof.

The proof is analogous to that of Lemma 6.7. If $E$ is $\sigma_{0}$-stable
there is nothing to prove, otherwise it must have at least a spherical stable
factor. Then one can write $\mathbf{v}=\mathbf{a}+\mathbf{b}$ with
$\mathbf{a}\in C_{\sigma_{0}}$ spherical, and $\mathbf{b}\in C_{\sigma_{0}}$.
By Lemma 6.8, the only spherical classes in $H$ are of the form
$\pm\mathbf{w}+n\mathbf{v}$ with $n\in\mathbb{Z}$; then $\mathbf{b}$ has to be
spherical as well, and there is only one integer $N$ such that
$\mathbf{a}\coloneqq\mathbf{w}+N\mathbf{v}$ and
$\mathbf{b}\coloneqq-\mathbf{w}+(1-N)\mathbf{v}$ are both
$\sigma_{0}$-effective. Moreover, $\mathbf{a}$ and $\mathbf{b}$ cannot be
expressed as the sum of other effective spherical classes. This implies that
the Jordan-Hölder filtration of $E$ is

$\epsilon\colon\quad B\to E\to A.$

Since $E$ is indecomposable, $(\epsilon)\neq 0$ in
$\operatorname{Ext}^{1}(A,B)\simeq\operatorname{Ext}^{1}(B,A)$, and we can
conclude as in Lemma 6.7. ∎

###### Proof of Theorem 6.4.

Suppose first that $\mathbf{v}$ is a spherical class. Proposition 6.2 shows
that up to a sign there exists a $\tau_{0}$-semistable sheaf $E$ which is
spherical and indecomposable. Since $\operatorname{Stab}^{\circ}(\mathcal{D})$
is connected and $\tau_{0}\in\operatorname{Stab}^{\circ}(\mathcal{D})$, there
is a path $\gamma$ of stability conditions in
$\operatorname{Stab}^{\circ}(\mathcal{D})$ connecting $\tau_{0}$ and $\sigma$.

Observe that the objects $E^{+}$ produced in Lemma 6.7 are in turn
indecomposable, because they are stable with respect to some stability
condition. Then, we can repeatedly apply Lemma 6.7 and conclude.

A similar argument, where one uses Lemma 6.9 instead of Lemma 6.7, works for
radical classes. ∎

###  6.3. Proof of Proposition 5.4

In this section, we prove that all stability conditions in
$\operatorname{Stab}^{\circ}(\mathcal{D})$ satisfy condition $(\ast)$ (see
Def. 5.3), i.e. that
$\operatorname{Stab}^{\circ}(\mathcal{D})=\operatorname{Stab}^{\dagger}(\mathcal{D})$.

It suffices to show that there does not exist a stability condition
$\sigma_{0}=(Z_{0},\mathcal{A}_{0})$ in
$\operatorname{Stab}^{\circ}(\mathcal{D})$ for which
$\operatorname{Im}\frac{Z(b)}{Z(a)}=0$. Suppose such $\sigma_{0}$ existed.
Acting with $\mathbb{C}$, we may assume that $Z_{0}(a),Z_{0}(b)\in\mathbb{R}$.
We further assume that $Z_{0}$ takes values in $\mathbb{Q}$. Then, choose
$x,y\in\mathbb{Z}$ coprime such that

(16) $xZ_{0}(a)+yZ_{0}(b)=0$

and $\mathbf{v}\coloneqq xa+yb$ is a positive radical vector. Thus,
$\mathbf{v}$ is a primitive radical vector with $Z_{0}(\mathbf{v})=0$. This
implies that there exists a neighborhood
$V\subset\operatorname{Stab}^{\circ}(\mathcal{D})$ of $\sigma_{0}$ such that
no $\sigma\in V$ admits semistable objects of class $\mathbf{v}$, since
semistability is a closed condition. But this contradicts Theorem 6.4.

If $Z_{0}$ takes values in $\mathbb{R}$, there may be no integer solutions to
(16), but for every $\epsilon>0$ there are integers $x,y$ such that

$\lvert xZ_{0}(a)+yZ_{0}(b)\rvert<\epsilon$

and $\mathbf{v}=xa+yb$ is a primitive radical vector. Choosing $\epsilon\ll
1$, the support property implies that there exists a neighborhood
$V\subset\operatorname{Stab}^{\circ}(\mathcal{D})$ of $\sigma_{0}$ such that
no $\sigma\in V$ admits semistable objects of class $\mathbf{v}$, and we
conclude in the same way.

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