}\phi^{5}_{\mathcal{L}}\coloneqq\phi_{\sigma}(l_{*}(\mathcal{L})).$

If $\mathcal{L}=\mathcal{O}_{C},$ then
$\phi_{1}=\phi_{\sigma}(i_{*}(\mathcal{O}_{C})),$
$\phi_{3}=\phi_{\sigma}(j_{*}(\mathcal{O}_{C})),$ and
$\phi_{5}=\phi_{\sigma}(l_{*}(\mathcal{O}_{C})).$

######  Remark 4.8.

Note that if $i_{*}(\mathbb{C}(x))$ is not $\sigma$-semistable, then
$\phi^{4}_{x}>\phi^{2}_{x}+1.$ Similarly, if $i_{*}(\mathcal{L})$ is not
$\sigma$-semistable, then $\phi^{5}_{\mathcal{L}}>\phi^{3}_{\mathcal{L}}+1.$

######  Remark 4.9.

After applying the Serre functor to the HN-filtration of $i_{*}(X)$, we obtain
the analogous results for $j_{*}(X)$ and $l_{*}(X).$ For example, if
$l_{*}(\mathbb{C}(x))$ is not $\sigma$-semistable, then
$\phi^{2}_{x}>\phi^{0}_{x}.$

######  Remark 4.10.

If $i_{*}(X)$ is strictly $\sigma$-semistable, where $X$ is either
$\mathbb{C}(x)$ or $\mathcal{L},$ then $j_{*}(X)$ and $l_{*}(X)$ are
$\sigma$-stable. Moreover, a Jordan-Hölder filtration is given by

$l_{*}(X)\rightarrow i_{*}(X)\rightarrow j_{*}(X)[1]\rightarrow l_{*}(X)[1].$

It now makes sense to define the following sets:

######  Definition 4.11.

We define the set $\Theta_{ij}$ of pre-stability conditions on
$\mathcal{T}_{C}$ for $ij=12,23$ or $31$ as follows:

$\displaystyle\Theta_{12}$ $\displaystyle=\\{$ $\displaystyle\sigma$
$\displaystyle\mid$ $\displaystyle
i_{*}(\mathbb{C}(x)),j_{*}(\mathbb{C}(x)),i_{*}(\mathcal{L}),j_{*}(\mathcal{L})\>\>\rm{stable}\textnormal{
for all line bundles $\mathcal{L}\in\operatorname{Coh}(C)$ and all $x\in C$ }$
$\displaystyle\\}$ , $\displaystyle\Theta_{23}$ $\displaystyle=\\{$
$\displaystyle\sigma$ $\displaystyle\mid$ $\displaystyle
j_{*}(\mathbb{C}(x)),l_{*}(\mathbb{C}(x)),j_{*}(\mathcal{L}),l_{*}(\mathcal{L})\>\>\rm{stable}\textnormal{
for all line bundles $\mathcal{L}\in\operatorname{Coh}(C)$ and all $x\in C$ }$
$\displaystyle\\}$ , $\displaystyle\Theta_{31}$ $\displaystyle=\\{$
$\displaystyle\sigma$ $\displaystyle\mid$ $\displaystyle
i_{*}(\mathbb{C}(x)),l_{*}(\mathbb{C}(x)),i_{*}(\mathcal{L}),l_{*}(\mathcal{L})\>\>\rm{stable}\textnormal{
for all line bundles $\mathcal{L}\in\operatorname{Coh}(C)$ and all $x\in C$ }$
$\displaystyle\\}$ .

Recall that we assumed that all the pre-stability conditions are locally
finite.

######  Remark 4.12.

Note that $\mathcal{S}_{\mathcal{T}_{C}}(\Theta_{12})=\Theta_{23}\textnormal{
and }\mathcal{S}_{\mathcal{T}_{C}}(\Theta_{23})=\Theta_{31}\textnormal{ and
}\mathcal{S}_{\mathcal{T}_{C}}(\Theta_{31})=\Theta_{12}.$

######  Theorem 4.13.

If $\sigma$ is a pre-stability condition on $\mathcal{T}_{C},$ then
$\sigma\in\Theta_{12}\cup\Theta_{23}\cup\Theta_{31}.$

######  Proof 9.

We first assume that $\sigma\notin\Theta_{23}$ and we prove that
$\sigma\in\Theta_{12}$ or $\sigma\in\Theta_{31}.$ Thus, there is a line bundle
$\mathcal{L}$ such that $j_{*}(\mathcal{L})$ or $l_{*}(\mathcal{L})$ is not
$\sigma$-stable, or either $j_{*}(\mathbb{C}(x))$ or $l_{*}(\mathbb{C}(x))$ is
not $\sigma$-stable for some $x\in C.$ Assume that there is $x\in C$ such that
$l_{*}(\mathbb{C}(x))$ is not $\sigma$-stable. We need to show that
$\sigma\in\Theta_{12},$ as it cannot be in $\Theta_{31}.$ By Remark 4.6, it
follows that $j_{*}(\mathbb{C}(x))$ and $i_{*}(\mathbb{C}(x))$ are
$\sigma$-stable. We will now show that for every line bundle $\mathcal{L}$ and
every $x\in C,$ we have that $j_{*}(X)$ and $i_{*}(X)$ are $\sigma$-stable,
where $X=\mathcal{L}$ or $X=\mathbb{C}(x).$

We prove this by contradiction, assume that there is a line bundle
$\mathcal{L},$ such that $i_{*}(\mathcal{L})$ is not $\sigma$-stable, which
implies that $j_{*}(\mathcal{L})$ and $l_{*}(\mathcal{L})$ are
$\sigma$-stable. Since $l_{*}(\mathbb{C}(x))$ is not $\sigma$-stable, by
Remark 4.7 we obtain that $\phi_{x}^{2}\geq\phi^{0}_{x}.$ Analogously, since
$i_{*}(\mathcal{L})$ is not $\sigma$-semistable, then we get
$\phi^{3}_{\mathcal{L}}+1\leq\phi^{5}_{\mathcal{L}}.$ Let us consider the non-
zero morphism $j_{*}(\mathbb{C}(x))\rightarrow j_{*}(\mathcal{L})[1].$ Since
by hypothesis both are stable and not isomorphic, we obtain that
$\phi^{2}_{x}<\phi^{3}_{\mathcal{L}}+1.$ We have a non-zero morphism
$l_{*}(\mathcal{L})[1]\rightarrow i_{*}(\mathbb{C}(x))[1],$ as both objects
are $\sigma$-stable and not isomorphic, we get
$\phi^{5}_{\mathcal{L}}<\phi^{0}_{x}.$ We now put all the inequalities
together, yielding

$\phi^{0}_{x}\leq\phi^{2}_{x}<\phi_{\mathcal{L}}^{3}+1\leq\phi^{5}_{\mathcal{L}}<\phi^{0}_{x},$

which is a contradiction. Therefore $i_{*}(\mathcal{L})$ has to be
$\sigma$-stable for all line bundles. Analogously we prove that
$j_{*}(\mathcal{L})$ has to be also stable for all line bundles $\mathcal{L}.$

We now assume that there is a point $y\in C,$ such that $j_{*}(\mathbb{C}(y))$
is not $\sigma$-stable. Then, by Remark 4.6 we obtain
$\phi^{0}_{y}-1\geq\phi_{y}^{4}.$ Note that
$[i_{*}(\mathbb{C}(x))]=[i_{*}(\mathbb{C}(y))]$ in the Grothendieck group. As
a consequence, we obtain $\phi^{0}_{x}=\phi^{0}_{y}+m,$ with $m\in\mathbb{Z}.$
But as $i_{*}(\mathcal{O}_{C})$ is $\sigma$-semistable and we have non-zero
morphisms from $i_{*}(\mathcal{O}_{C})$ to $i_{*}(\mathbb{C}(x))$ and to
$i_{*}(\mathbb{C}(y),$ we also get non-zero morphisms from
$i_{*}(\mathbb{C}(x))$ and $i_{*}(\mathbb{C}(y))$ to
$i_{*}(\mathcal{O}_{C})[1],$ we obtain
$\phi_{1}<\phi^{0}_{y}<\phi_{1}+1\textnormal{ and
}\phi_{1}<\phi^{0}_{x}<\phi_{1}+1,$ which implies that
$\phi^{0}_{y}=\phi^{0}_{x}.$

Since we have a non-zero morphism $j_{*}(\mathcal{O}_{C})\rightarrow
l_{*}(\mathbb{C}(y)),$ we obtain $\phi_{3}<\phi^{4}_{y}$ and from the morphism
$j_{*}(\mathbb{C}(x))\rightarrow j_{*}(\mathcal{O}_{C})[1]$ we obtain
$\phi^{2}_{x}<\phi_{3}+1.$

As a consequence, we get
$\phi_{3}+1<\phi^{4}_{y}+1\leq\phi^{0}_{y}=\phi^{0}_{x}\leq\phi^{2}_{x}<\phi_{3}+1,$
which is a contradiction. Therefore, we obtain that $j_{*}(\mathbb{C}(y))$ is
$\sigma$-stable. Then $\sigma\in\Theta_{12}.$ The other cases follow
analogously.

######  Corollary 4.14.

Let $\sigma$ be a pre-stability condition on $\mathcal{T}_{C},$ with
$i_{*}(\mathbb{C}(x))$ $\sigma$-stable for $x\in C$. Then, for all $y\in C,$
we have that $i_{*}(\mathbb{C}(y))$ is $\sigma$-stable and
$\phi^{0}_{x}=\phi^{0}_{y}.$

######  Proof 10.

By Theorem 4.13, we have that if $\sigma\in\Theta_{12}$ or
$\sigma\in\Theta_{31},$ it follows directly that for all $y\in C$ we have that
$i_{*}(\mathbb{C}(y))$ is $\sigma$-stable. Note that
$[i_{*}(\mathbb{C}(x))]=[i_{*}(\mathbb{C}(y))]$ in the Grothendieck group.
This implies that $\phi^{0}_{x}=\phi^{0}_{y}+m,$ with $m\in\mathbb{Z}.$ By
stability, we obtain $m=0.$

######  Remark 4.15.

Analogously, by using Serre duality, we prove Corollary 4.14 for
$j_{*}(\mathbb{C}(x))$ and $l_{*}(\mathbb{C}(x)).$

By Corollary 4.14, if $i_{*}(\mathbb{C}(x))$ is $\sigma$-stable for some $x\in
C,$ then $\phi^{0}_{x}$ does not depend on $x.$ Therefore, we define
$\phi_{0}\coloneqq\phi^{0}_{x}$, $\phi_{2}\coloneqq\phi^{2}_{x}$,
$\phi_{4}\coloneqq\phi^{4}_{x}.$

######  Lemma 4.16.

Let $\sigma=(Z_{r},\mathcal{A}_{r})$ be a pre-stability condition constructed
in Lemma 3.48. Then

  1. (1)

$i_{*}(\mathbb{C}(x))$ $l_{*}(\mathbb{C}(x))$ and $j_{*}(\mathbb{C}(x))$ are
stable.

  2. (2)

$i_{*}(\mathcal{O}_{C})$ and $j_{*}(\mathcal{O}_{C})$ are $\sigma$-stable if
and only if $\phi_{1}<\frac{3}{2}.$

  3. (3)

$i_{*}(\mathcal{O}_{C}),$ $j_{*}(\mathcal{O}_{C})$ and
$l_{*}(\mathcal{O}_{C})$ are $\sigma$-stable if and only if
$\frac{1}{2}=\phi_{3}<\phi_{5}<\phi_{1}<\frac{3}{2}.$

  4. (4)

$j_{*}(\mathcal{O}_{C})$ and $l_{*}(\mathcal{O}_{C})$ are $\sigma$-stable and
$i_{*}(\mathcal{O}_{C})$ is not $\sigma$-stable if and only if
$\phi_{1}\geq\frac{3}{2}$ and $\phi_{5}>\frac{1}{2}.$ Moreover, we have that
$i_{*}(\mathcal{O}_{C})$ and $l_{*}(\mathcal{O}_{C})$ are $\sigma$-stable and
$j_{*}(\mathcal{O}_{C})$ is not $\sigma$-stable if and only if
$\phi_{1}\geq\frac{3}{2}$ and $\phi_{5}\leq\frac{1}{2}.$

######  Proof 11.

We start by proving the first part. By Remark 3.49, we have that
$i_{*}(\mathbb{C}(x))[-1],$ $l_{*}(\mathbb{C}(x))[-1]$ and
$j_{*}(\mathbb{C}(x))$ are in $\mathcal{A}_{r}$ and that
$j_{*}(\mathbb{C}(x))$ is stable of phase one. We will now show that
$i_{*}(\mathbb{C}(x))[-1]$ is $\sigma$-stable. By contradiction we first
assume that $i_{*}(\mathbb{C}(x))$ is not $\sigma$-stable. As a consequence of
Remark 4.9, we have that $l_{*}(\mathbb{C}(x))$ is $\sigma$-stable and
$\phi_{4}\geq\phi_{2}+1,$ but $\phi_{2}=1$ and $1<\phi_{4}<2,$ which gives a
contradiction. The same reasoning works to prove that
$l_{*}(\mathbb{C}(x))[-1]$ is stable. For the second part, note that
$\phi_{1}=\phi(i_{*}(\mathcal{O}_{C}))$ and
$\phi_{3}=\phi(j_{*}(\mathcal{O}_{C}))=\frac{1}{2}$ makes sense, as Lemma 3.50
implies that $j_{*}(\mathcal{O}_{C})\in\mathcal{A}_{r}$ and that
$i_{*}(\mathcal{O}_{C})$ is in $\mathcal{A}_{r}$ or in $\mathcal{A}_{r}[1].$
Moreover, if $i_{*}(\mathcal{O}_{C})$ and $j_{*}(\mathcal{O}_{C})$ are
$\sigma$-stable, then $\phi_{1}<\phi_{3}+1=\frac{3}{2},$ because there is a
non-zero morphism $i_{*}(\mathcal{O}_{C})\rightarrow
j_{*}(\mathcal{O}_{C})[1].$ We will now prove the other direction. We assume
that $\phi_{1}<\frac{3}{2}$ and that $i_{*}(\mathcal{O}_{C})$ is not stable.
By Remark 4.9, the statement follows. The other two statements follow in a
similar way.

###  4.1. Pre-stability conditions in $\Theta_{12}$

We are going to show that pre-stability conditions $\sigma\in\Theta_{12}$ are
given as the ones constructed in Corollary 3.36 or in Lemma 3.48.

We first characterise the hearts of the pre-stability conditions in terms of
the stability of the skyscraper sheaves. We study pre-stability conditions
satisfying that $j_{*}(\mathbb{C}(x))$ is stable of phase one. We separate
them into two cases when $l_{*}(\mathbb{C}(x))$ is not $\sigma$-stable and
when it is. If $l_{*}(\mathbb{C}(x))$ is not $\sigma$-stable, then the pre-
stability conditions are CP-glued. If $l_{*}(\mathbb{C}(x))$ is
$\sigma$-stable, we obtain stability conditions of the form of Lemma 3.47. We
follow closely [12, Proposition 10.1].

######  Lemma 4.17.

Let $\sigma=(Z,\mathcal{A})$ be a pre-stability condition such that
$i_{*}(\mathbb{C}(x))$ is $\sigma$-stable, $j_{*}(\mathbb{C}(x))$ is
$\sigma$-stable of phase one and $l_{*}(\mathbb{C}(x))$ is not
$\sigma$-stable.

We assume that $i_{*}(\mathbb{C}(x))[n]\in\mathcal{A}.$ If
$E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{A},$ then $H^{i}(E_{1})=0,$
unless $i=-n-1,-n.$ Also $H^{i}(E_{2})=0$ unless $i=0$ and
$H^{i}(C(\varphi))=0,$ unless $i=-n-1,-n-2,0.$

######  Proof 12.

First note that $n\geq 0.$ Indeed, as $l_{*}(\mathbb{C}(x))$ is not
$\sigma$-stable, it implies that $\phi_{0}-\phi_{2}\leq 0.$ As $\phi_{2}=1$ we
have $\phi_{0}\leq 1,$ also $0<\phi(i_{*}(\mathbb{C}(x))[n])=\phi_{0}+n\leq
1.$ Combining the inequalities above, we get $n\geq 0.$

Let $E\in\mathcal{A}$ be stable with phase $0<\phi(E)<1.$ As
$j_{*}(\mathbb{C}(x))$ is stable we have that
$\operatorname{Hom}_{\mathcal{T}_{C}}^{i}(E,j_{*}(\mathbb{C}(x)))=0,$ for all
$i<0.$ By adjointness, we have

$0=\operatorname{Hom}_{\mathcal{T}_{C}}^{i}(E,j_{*}(\mathbb{C}(x)))=\operatorname{Hom}^{i}_{D^{b}(C)}(C(\varphi),\mathbb{C}(x)).$

Moreover, by stability we have
$\operatorname{Hom}^{i}(i_{*}(\mathbb{C}(x))[n],E)=0,$ for all $i<0.$ By
adjointness and Serre duality in $D^{b}(C),$ we get

$0=\operatorname{Hom}_{\mathcal{T}_{C}}^{i}(i_{*}(\mathbb{C}(x))[n],E)=\operatorname{Hom}_{D^{b}(C)}^{i}(\mathbb{C}(x)[n],C(\varphi)[-1])=\operatorname{Hom}^{n-i+2}_{D^{b}(C)}(C(\varphi),\mathbb{C}(x))^{*},$

i.e. $\operatorname{Hom}_{D^{b}(C)}^{j}(C(\varphi),\mathbb{C}(x))=0$ for
$j>n+2.$ Therefore, we obtain
$\operatorname{Hom}^{i}(C(\varphi),\mathbb{C}(x))=0,$ unless $0\leq i\leq
n+2.$ By [13, Proposition 5.4], it follows that $C(\varphi)$ is isomorphic to
a complex of locally-free sheaves and $H^{i}(C(\varphi))=0\textnormal{ unless
}-n-2\leq i\leq 0.$

Similarly, by stability
$\operatorname{Hom}_{\mathcal{T}_{C}}^{i}(E,i_{*}(\mathbb{C}(x))[n])=0,$ for
all $i<0.$ By adjointness, we get
$0=\operatorname{Hom}_{\mathcal{T}_{C}}^{i}(E,i_{*}(\mathbb{C}(x))[n])=\operatorname{Hom}^{i+n}_{D^{b}(C)}(E_{1},\mathbb{C}(x)),$
i.e. $\operatorname{Hom}_{D^{b}(C)}^{j}(E_{1},\mathbb{C}(x))=0$ for $j<n.$
Since $l_{*}(\mathbb{C}(x))$ is not $\sigma$-semistable, it follows from its
HN-filtration that $\phi_{2}\geq\phi_{4}\geq\phi_{0},$ which implies that
$l_{*}(\mathbb{C}(x))[n]\in\mathcal{P}(0,n+1],$ because
$j_{*}(\mathbb{C}(x))\in\mathcal{P}(0,1]$ and
$i_{*}(\mathbb{C}(x))[n]\in\mathcal{P}(0,1].$ As a consequence, we have
$\operatorname{Hom}^{i}(l_{*}(\mathbb{C}(x))[n],E)=0,$ for all $i<0.$ By
adjointness and Serre duality in $D^{b}(C),$ we get
$0=\operatorname{Hom}_{T_{C}}^{i}(l_{*}(\mathbb{C}(x))[n],E)=\operatorname{Hom}^{n+1-i}_{D^{b}(C)}(E_{1},\mathbb{C}(x))^{*},$
i.e. $\operatorname{Hom}_{D^{b}(C)}^{j}(E_{1},\mathbb{C}(x))=0$ for $j>n+1.$
Consequently, we obtain $\operatorname{Hom}^{i}(E_{1},\mathbb{C}(x))=0$ unless
$n\leq i\leq n+1.$ By [13, Proposition 5.4], we have that $E_{1}$ is
isomorphic to a length two complex of locally-free sheaves and
$H^{i}(E_{1})=0\textnormal{ unless }i=-n,-n-1.$

We will now prove a similar result for $E_{2}.$ In this case we use that
$j_{*}(\mathbb{C}(x))$ is stable of phase one. First of all, by the same
reasoning as above, $l_{*}(\mathbb{C}(x))\in\mathcal{P}(-n,1].$ Hence, we get
$\operatorname{Hom}^{i}(E,l_{*}(\mathbb{C}(x)))=0,$ for all $i<0.$ By
adjointness, we obtain
$0=\operatorname{Hom}_{\mathcal{T}_{C}}^{i}(E,l_{*}(\mathbb{C}(x)))=\operatorname{Hom}^{i}_{D^{b}(C)}(E_{2},\mathbb{C}(x)).$
In addition, since $j_{*}(\mathbb{C}(x))$ is stable of phase $1,$ we get
$\operatorname{Hom}_{\mathcal{T}_{C}}^{i}(j_{*}(\mathbb{C}(x)),E)=0$ for
$i\leq 0.$ By adjointness and Serre duality in $D^{b}(C),$ we have
$0=\operatorname{Hom}_{\mathcal{T}_{C}}^{i}(j_{*}(\mathbb{C}(x)),E)=\operatorname{Hom}^{1-i}_{D^{b}(C)}(E_{2},\mathbb{C}(x))^{*},$
i.e. $\operatorname{Hom}_{D^{b}(C)}^{j}(E_{2},\mathbb{C}(x))=0$ for $j>0.$
Thus, we obtain $\operatorname{Hom}^{i}(E_{2},\mathbb{C}(x))=0,$ unless $i=0.$
By [13, Proposition 5.4], we have that $E_{2}$ is isomorphic to a length one
complex of locally-free sheaves and $H^{i}(E_{2})=0\textnormal{ unless }i=0.$

The triangle $E_{1}\rightarrow E_{2}\rightarrow C(\varphi)$ induces a long
exact sequence in cohomology. As a consequence, we have that
$H^{i}(C(\varphi))=0\textnormal{ unless }-n-2,-n-1,0.$

If $E\ncong j_{*}(\mathbb{C}(x))$ is $\sigma$-stable with phase one, then
$H^{i}(E_{1})=0$ unless $i=-n-1\textnormal{ and }$$H^{i}(E_{2})=0\textnormal{
for all }i.$ This implies that $H^{i}(C(\varphi))=0\textnormal{ unless
}i=-n-2$ and that $E_{1}$ is torsion free.

######  Lemma 4.18.

Let $\sigma=(Z,\mathcal{A})$ be a pre-stability condition such that
$l_{*}(\mathbb{C}(x)),i_{*}(\mathbb{C}(x))$ are $\sigma$-stable and
$j_{*}(\mathbb{C}(x))$ is $\sigma$-stable of phase one. Then, for
$E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{T}_{C}$ we have

  1. (1)

If $E\in\mathcal{A},$ then $H^{i}(E_{j})=0,$ unless $i={0,1},$ for $j=1,2.$
Also $H^{i}(C(\varphi))=0,$ unless $i={-1,0}.$ Moreover,
$H^{-1}(C(\varphi)),H^{0}(E_{1})$ are torsion-free.

  2. (2)

If $E$ is stable of phase one, then either $E=j_{*}(T),$ where
$T\in\operatorname{Coh}(C)$ a torsion sheaf, or $E\in\operatorname{TCoh}(C)$
with $H^{0}(C(\varphi))=0.$ We have that $E_{1}$ and $E_{2}$ are torsion-free.

  3. (3)

$\operatorname{TCoh}(C)\subseteq\mathcal{P}(0,2]$

  4. (4)

The pair $\mathcal{T}=\operatorname{TCoh}(C)\cap\mathcal{P}(1,2]\textnormal{
and }\mathcal{F}=\operatorname{TCoh}(C)\cap\mathcal{P}(0,1]$ defines a torsion
pair on $\operatorname{TCoh}(C).$ Moreover, the heart $\mathcal{A}$ is the
corresponding tilt.

######  Proof 13.

First note that
$i_{*}(\mathbb{C}(x))[-1],l_{*}(\mathbb{C}(x))[-1]\in\mathcal{A}.$ Indeed,
there are non-zero morphisms $i_{*}(\mathbb{C}(x))\rightarrow
j_{*}(\mathbb{C}(x)[1]),$ $j_{*}(\mathbb{C}(x))\rightarrow
l_{*}(\mathbb{C}(x))$ and $l_{*}(\mathbb{C}(x))\rightarrow
i_{*}(\mathbb{C}(x)).$ By stability, it follows that $1<\phi_{4}<\phi_{0}<2.$

We start by proving part one. Let $E\in\mathcal{A}$ be stable with phase
$0<\phi(E)<1.$ As $j_{*}(\mathbb{C}(x))$ is stable we have that
$\operatorname{Hom}_{\mathcal{T}_{C}}^{i}(E,j_{*}(\mathbb{C}(x)))=0,$ for all
$i<0.$ By adjointness, we have
$0=\operatorname{Hom}_{\mathcal{T}_{C}}^{i}(E,j_{*}(\mathbb{C}(x)))=\operatorname{Hom}^{i}_{D^{b}(C)}(C(\varphi),\mathbb{C}(x)).$
Moreover, by stability we have
$\operatorname{Hom}^{i}(i_{*}(\mathbb{C}(x))[-1],E)=0,$ for all $i<0.$ By
adjointness and Serre duality in $D^{b}(C),$ we get
$0=\operatorname{Hom}_{T_{C}}^{i}(i_{*}(\mathbb{C}(x))[-1],E)=\operatorname{Hom}^{1-i}_{D^{b}(C)}(C(\varphi),\mathbb{C}(x))^{*},$
i.e. $\operatorname{Hom}_{D^{b}(C)}^{j}(C(\varphi),\mathbb{C}(x))=0$ for
$j>1.$ Therefore, we obtain
$\operatorname{Hom}^{i}(C(\varphi),\mathbb{C}(x))=0,$ unless $0\leq i\leq 1.$
By [13, Proposition 5.4], it follows that $C(\varphi)$ is isomorphic to a
length two complex of locally-free sheaves and
$H^{i}(C(\varphi))=0\textnormal{ unless }i={-1,0}.$

Similarly, we obtain $\operatorname{Hom}^{i}(E_{1},\mathbb{C}(x))=0$ unless
$-1\leq i\leq 0.$ By [13, Proposition 5.4], we have that $E_{1}$ is isomorphic
to a length two complex of locally-free sheaves and
$H^{i}(E_{1})=0\textnormal{ unless }i={0,1}.$

We prove a similar result for $E_{2}.$ In this case we use that
$j_{*}(\mathbb{C}(x))$ is stable of phase one. We obtain
$\operatorname{Hom}^{i}(E_{2},\mathbb{C}(x))=0,$ unless $-1\leq i\leq 0.$ By
[13, Proposition 5.4], we have that $E_{2}$ is isomorphic to a length two
complex of locally-free sheaves and $H^{i}(E_{2})=0\textnormal{ unless
}i={0,1}.$ This concludes the proof of the first part.

We will now proceed to proving the second part. Let $E\in\mathcal{P}(1)$ be a
stable object, which is not isomorphic to $j_{*}(T),$ where $T$ is a torsion
sheaf. Since the phase of $E$ is one, we have $H^{i}(E_{1})=0\textnormal{
unless }i={0}\textnormal{ and }H^{i}(E_{2})=0\textnormal{ unless }i={0}.$

For the third part, we assume that $E\in\operatorname{TCoh}(C).$ If
$F\in\mathcal{P}((2,\infty)),$ then by the first part $F\in
D^{\leq-2}\subseteq D^{\leq-1},$ where $(D^{\leq 0},D^{\geq 0})$ is the
standard t-structure. Consequently, we have
$0=\operatorname{Hom}_{\mathcal{T}_{C}}(D^{\leq 0},D^{\geq
1})=\operatorname{Hom}(D^{\leq-1},D^{\geq 0}),$ therefore
$\operatorname{Hom}_{\mathcal{T}_{C}}(F,E)=0.$ Analogously, we have that if
$B\in\mathcal{P}(\leq 0),$ then $B\in D^{\geq 1}.$ As
$\operatorname{Hom}_{\mathcal{T}_{C}}(D^{\leq 0},D^{\geq 1})=0,$ we obtain
$\operatorname{Hom}_{\mathcal{T}_{C}}(E,B)=0.$ It follows that
$E\in\mathcal{P}(0,2].$

We will now prove the fourth part. Let $E\in\operatorname{TCoh}(C).$ By the
third part of the statement, there is a triangle $A\rightarrow E\rightarrow
B\rightarrow A[1],$ where $A\in\mathcal{P}(1,2]$ and $B\in\mathcal{P}(0,1].$
After applying $i^{*},$ we obtain a long exact sequence of cohomology and by
part one we obtain $H^{-1}(i^{*}(A))=H^{1}(i^{*}(B))=0.$ This implies that
$i^{*}(A),i^{*}(B)\in\operatorname{Coh}(C).$ Analogously, we have
$j^{!}(A),j^{!}(B)\in\operatorname{Coh}(C)$ and we obtain
$A,B\in\operatorname{TCoh}(C).$

Moreover, if $A\in\mathcal{T},$ we have additional information. Indeed, as we
have $H^{i}(j^{*}(B))=0$ unless $i=-1,0$ and $H^{i}(j^{*}(A))=0$ unless
$i=-2,-1.$ After taking the long exact sequence we obtain $H^{i}(j^{*}(A))=0$
unless $i=-1.$ As a consequence, if $A=A_{1}\xrightarrow{g}A_{2},$ it follows
that $\operatorname{Coker}(g)=0.$

By applying Serre duality of doing exactly the same proof under the new
hypothesis, we obtain the following results.

######  Lemma 4.19.

Let $\sigma=(Z,\mathcal{A})$ be a pre-stability condition such that
$j_{*}(\mathbb{C}(x)),i_{*}(\mathbb{C}(x))$ are $\sigma$-stable and
$i_{*}(\mathbb{C}(x))$ is $\sigma$-stable of phase one. Then, for
$E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{T}_{C}$ we have

  1. (1)

If $E\in\mathcal{A},$ then $H^{i}(E_{j})=0,$ unless $i={-1,0},$ for $j=1,2.$
Also $H^{i}(C(\varphi))=0,$ unless $i={-1,0}.$ We also have that
$H^{-1}(E_{1}),H^{-1}(E_{2})$ are torsion-free.

  2. (2)

If $E$ is stable of phase one, then either $E=i_{*}(T),$ where
$T\in\operatorname{Coh}(C)$ is a torsion sheaf, or $E\in\mathcal{H}_{31}$ with
$H^{0}(E_{1})=0.$ Moreover, we have that $E_{2}$ and $C(\varphi)$ are torsion-
free.

  3. (3)

$\mathcal{H}_{31}\subseteq\mathcal{P}(0,2]$

  4. (4)

The pair $\mathcal{T}=\mathcal{H}_{31}\cap\mathcal{P}(1,2]\textnormal{ and
}\mathcal{F}=\mathcal{H}_{31}\cap\mathcal{P}(0,1]$ defines a torsion pair on
$\mathcal{H}_{31}.$ Moreover, the heart $\mathcal{A}$ is the corresponding
tilt.

######  Lemma 4.20.

Let $\sigma=(Z,\mathcal{A})$ be a pre-stability condition such that
$i_{*}(\mathbb{C}(x)),j_{*}(\mathbb{C}(x))$ are $\sigma$-stable and
$l_{*}(\mathbb{C}(x))$ is $\sigma$-stable of phase one. Then, for
$E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{T}_{C}$ we have

  1. (1)

If $E\in\mathcal{A},$ then $H^{i}(E_{1})=0,$ unless $i={0,1}$ and
$H^{i}(E_{2})=0,$ unless $i={0,-1}.$ We also obtain that
$H^{i}(C(\varphi))=0,$ unless $i={-1,0}.$ We also have that
$H^{-1}(C(\varphi)),H^{-1}(E_{2})$ are torsion-free.

  2. (2)

If $E$ is stable of phase one, then either $E=l_{*}(T),$ where
$T\in\operatorname{Coh}(C)$ a torsion sheaf, or $E\in\mathcal{H}_{23}$ with
$H^{0}(E_{2})=0$ and $H^{0}(E_{1})$ torsion-free sheaves.

  3. (3)

$\mathcal{H}_{23}\subseteq\mathcal{P}(0,2]$

  4. (4)

The pair $\mathcal{T}=\mathcal{H}_{23}\cap\mathcal{P}(1,2]\textnormal{ and
}\mathcal{F}=\mathcal{H}_{23}\cap\mathcal{P}(0,1]$defines a torsion pair on
$\mathcal{H}_{23}.$ Moreover, the heart $\mathcal{A}$ is the corresponding
tilt.

We will now study the orbit of $\sigma\in\Theta_{12}$ under the action of
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ in order to choose a simpler
representative.

######  Proposition 4.21.

For every $\sigma\in\Theta_{12},$ there is
$g\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ such that for $\sigma
g=(Z,\mathcal{A})$ we can find stability conditions
$\sigma_{1}=(Z_{1},\operatorname{Coh}^{r}(C))\textnormal{ and
}\sigma_{2}=(Z_{\mu},\operatorname{Coh}(C))\in\operatorname{Stab}(C)$ with
$\operatorname{Coh}^{r}_{1}(C)\subseteq\mathcal{A}$ and $r>-1,$
$\operatorname{Coh}_{2}(C)\subseteq\mathcal{A},$
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}_{1}}}=Z_{1}$ and
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}_{2}}}=Z_{\mu}.$

######  Proof 14.

By the stability of $i_{*}(\mathbb{C}(x))$ and $i_{*}(\mathcal{O}_{C}),$ we
have $\phi_{1}<\phi_{0}<\phi_{1}+1.$ Therefore, there is an orientation
preserving transformation $M\colon\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$
satisfying that $(A,D)\mapsto(-1,0)$ and $(B,C)\mapsto(0,1),$ where
$Z(i_{*}(\mathbb{C}(x))=A+Di$ and $Z(i_{*}(\mathcal{O}_{C}))=B+Ci.$ There is
an increasing function $f\colon\mathbb{R}\rightarrow\mathbb{R}$ that satisfies
$f(x+1)=f(x)+1$ with $f(1)=\phi_{0},$ $f(1/2)=\phi_{1}.$ We obtain
$(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$. The stability
condition $\sigma^{{}^{\prime}}\coloneqq\sigma(T,f)$ satisfies

$Z^{{}^{\prime}}(r_{1},d_{1},0,0)=-d_{1}+r_{1}i\textnormal{ and
}\operatorname{Coh}_{1}(C)\subseteq\mathcal{A}^{{}^{\prime}},$

where $\sigma^{{}^{\prime}}=(Z^{{}^{\prime}},\mathcal{A}^{{}^{\prime}}).$
Indeed, we have

$\displaystyle i_{*}(\mathbb{C}(x))\in\mathcal{P}(\phi_{0})$ $\displaystyle=$
$\displaystyle\mathcal{P}^{{}^{\prime}}(1),$ $\displaystyle
i_{*}(\mathcal{O}_{C})\in\mathcal{P}(\phi_{1})$ $\displaystyle=$
$\displaystyle\mathcal{P}^{{}^{\prime}}(1/2),$ $\displaystyle
i_{*}(\mathcal{L})\in\mathcal{P}(\phi_{\mathcal{L}})$ $\displaystyle=$
$\displaystyle\mathcal{P}^{{}^{\prime}}(t_{\mathcal{L}}),$

with $t_{\mathcal{L}}\in(0,1).$ Therefore, all skyscraper sheaves and line
bundles in $\operatorname{Coh}_{1}(C)$ are in
$\mathcal{P}^{{}^{\prime}}(0,1].$ Since any object in
$\operatorname{Coh}_{1}(C)$ admits a filtration with quotients either
isomorphic to skyscraper sheaves or to line bundles, we obtain
$\operatorname{Coh}_{1}(C)\subseteq\mathcal{A}^{{}^{\prime}}.$

As the action of $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ on the set of
pre-stability conditions preserves the stability of the objects and
$j_{*}(\mathbb{C}(x)),$ $j_{*}(\mathcal{O}_{C})$ are $\sigma$-stable, we have
$\phi_{3}<\phi_{2}<\phi_{3}+1$ in $\sigma^{{}^{\prime}}.$ Therefore, we can
find $g_{1}=(T_{1},f_{1})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$
such that $\sigma^{{}^{\prime\prime}}\coloneqq\sigma^{\prime}(T_{1},f_{1})$
satisfies
${\left.\kern-1.2ptZ^{{}^{\prime\prime}}\vphantom{\big{|}}\right|_{\mathcal{C}}}_{2}=Z_{\mu}$
and $\operatorname{Coh}_{2}(C)\subseteq\mathcal{A}^{{}^{\prime\prime}}.$

Moreover, if we consider
$\sigma_{1}=(Z_{1},\operatorname{Coh}^{r}(C))\in\operatorname{Stab}(C)$ with
$f_{1}(0)=r\in\mathbb{R}$, which under our correspondence with
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ is
$(T_{1},f_{1})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ then
$\sigma^{{}^{\prime\prime}}$ also satisfies that
${\left.\kern-1.2ptZ^{{}^{\prime\prime}}\vphantom{\big{|}}\right|_{\mathcal{C}}}_{1}=Z_{1}$
and $\operatorname{Coh}_{1}^{r}(C)\subseteq\mathcal{A}^{{}^{\prime\prime}}.$
Indeed, by definition $Z^{{}^{\prime\prime}}=T_{1}^{-1}\circ Z^{{}^{\prime}}.$
Consequently, we can assert that
${\left.\kern-1.2ptZ^{{}^{\prime\prime}}\vphantom{\big{|}}\right|_{\mathcal{C}}}_{1}=T_{1}^{-1}\circ
Z_{\mu}.$

We will now show that
$\operatorname{Coh}_{1}^{r}(C)\subseteq\mathcal{A}^{{}^{\prime\prime}},$ where
$f_{1}(0)=r=n+\theta,$ and $n\in\mathbb{Z}$ and $\theta\in[0,1).$ We prove
this in several steps. We first show that
$\operatorname{Coh}_{1}(C)[n]\subseteq\mathcal{P}(-1,1].$ Then, we construct a
torsion pair of $\operatorname{Coh}_{1}(C)[n]$ and we compare it with
$\operatorname{Coh}(C)[n]=\langle\mathcal{T}_{\theta}[n],\mathcal{F}_{\theta}[n]\rangle,$
which is the torsion pair given in Remark 2.52.

We first show that
$i_{*}(\mathbb{C}(x)[n])\in\mathcal{A}^{{}^{\prime\prime}}.$ Note that
$f_{1}(\phi_{0})=1,$ because

$i_{*}(\mathbb{C}(x))\in\mathcal{P}^{{}^{\prime}}(1)=\mathcal{P}^{{}^{\prime\prime}}(f_{1}^{-1}(1))=\mathcal{P}^{{}^{\prime\prime}}(\phi_{0})\textnormal{
and }f_{1}(-n)=\theta.$

We apply $f_{1}^{-1}$ to the following inequality $\theta<1\leq\theta+1$ and
we obtain

(4.6) $-n<\phi_{0}\leq-n+1,$

which is equivalent to
$i_{*}(\mathbb{C}(x)[n])\in\mathcal{A}^{{}^{\prime\prime}}.$ By stability
$-n<2.$

######  Claim 4.22.

$f_{1}(0)=r>-1.$

######  Proof 15.

Since $r=n+\theta$ and $n\geq-1,$ we obtain $r\geq-1.$ We just need to prove
that $r\neq-1.$ Assume that $r=-1.$ Then if
$Z_{1}(r_{1},d_{1})=A_{1}d_{1}+Br_{1}+(C_{1}r_{1}+D_{1}d_{1})i,$ we get
$D_{1}=0.$ By the stability of $i_{*}(\mathbb{C}(x))$ and
$j_{*}(\mathbb{C}(x))$ in $\sigma^{{}^{\prime\prime}},$ we get $\phi_{0}<2.$
But, by the definition of $Z_{1},$ we obtain
$\phi_{0}=\phi(i_{*}(\mathbb{C}(x))[-1])+1=2,$ which is a contradiction.

We will now study line bundles $\mathcal{L}.$ Note that
$f_{1}(\phi_{\mathcal{L}})=t_{\mathcal{L}},$ with $t_{\mathcal{L}}\in(0,1)$
because

$i_{*}(\mathcal{L})\in\mathcal{P}^{{}^{\prime}}(t_{\mathcal{L}})=\mathcal{P}^{{}^{\prime\prime}}(f^{-1}(t_{\mathcal{L}}))=\mathcal{P}^{{}^{\prime\prime}}(\phi_{\mathcal{L}}).$

The proof naturally falls into two cases:

Case 1: $\theta<t_{\mathcal{L}}\leq\theta+1.$ We apply $f_{1}^{-1}$ to the
inequality above and we obtain $-n<\phi_{\mathcal{L}}\leq-n+1,$ which is
equivalent to $i_{*}(\mathcal{L}[n])\in\mathcal{A}^{{}^{\prime\prime}}.$

Case 2: $\theta-1<t_{\mathcal{L}}\leq\theta.$ We apply $f_{1}^{-1}$ to the
inequality above and we obtain $-n-1<\phi_{\mathcal{L}}\leq-n,$ which is
equivalent to $i_{*}(\mathcal{L}[n+1])\in\mathcal{A}^{{}^{\prime\prime}}.$

######  Claim 4.23.

$\operatorname{Coh}_{1}(C)\subseteq\mathcal{P}(-1,1].$

######  Proof 16.

We just proved that all line bundles and the skyscraper sheaves are in
$\mathcal{P}(-n-1,-n+1],$ therefore
$\operatorname{Coh}_{1}(C)[n]\subseteq\mathcal{P}(-1,1].$

We set
$\mathcal{T}_{1}=\operatorname{Coh}_{1}(C)[n]\cap\mathcal{P}(0,1]\textnormal{
and }\mathcal{F}_{1}=\operatorname{Coh}_{1}(C)[n]\cap\mathcal{P}(-1,0].$

######  Claim 4.24.

$(\mathcal{T}_{1},\mathcal{F}_{1})$ is a torsion pair of
$\operatorname{Coh}_{1}(C)[n].$

######  Proof 17.

The proof falls naturally into two cases:

Case 1: $l_{*}(\mathbb{C}(x))$ is not $\sigma^{{}^{\prime\prime}}$-stable.

Let $E\in\operatorname{Coh}(C).$ We have $E[n]\in\operatorname{Coh}(C)[n].$
Since $\operatorname{Coh}_{1}(C)[n]\subseteq\mathcal{P}(-1,1],$ there are
objects $T\in\mathcal{P}(0,1]$ and $F\in\mathcal{P}(-1,1]$ such that
$0\rightarrow T\rightarrow i_{*}(E)[n]\rightarrow F\rightarrow 0.$

We obtain long exact sequences in cohomology and by Lemma 4.17, we have that
$H^{i}(T_{1})=H^{i}(F_{1})=0$ unless $i=-n$ and that $T_{2}=F_{2}=0.$ As a
consequence, we get
$\operatorname{Coh}_{1}(C)[n]=(\mathcal{T}_{1},\mathcal{F}_{1})$ and
$\mathcal{T}_{1},\mathcal{F}_{1}[1]\subseteq\mathcal{A}^{{}^{\prime\prime}}.$

Case 2: $l_{*}(\mathbb{C}(x))$ is $\sigma^{{}^{\prime\prime}}$-stable.

Note that the stability of $l_{*}(\mathbb{C}(x))$ implies, as proved in Lemma
4.18, that $n=-1$ and that $l_{*}(\mathbb{C}(x))[-1]\in
A^{{}^{\prime\prime}}.$

Let $E\in\operatorname{Coh}(C).$ We have $E[-1]\in\operatorname{Coh}(C)[-1].$
Since $\operatorname{Coh}_{1}(C)[-1]\in\mathcal{P}(-1,1],$ there are objects
$T\in\mathcal{P}(0,1]$ and $F\in\mathcal{P}(-1,1]$ such that $0\rightarrow
T\rightarrow i_{*}(E)[-1]\rightarrow F\rightarrow 0.$ We obtain long exact
sequences in cohomology. By Lemma 4.18, we have $H^{i}(T_{1})=0$ unless
$i=0,1$ and $H^{i}(F_{1})=0$ unless $i=1,2.$ Then, we obtain that
$H^{i}(T_{1})=H^{i}(F_{1})=0$ unless $i=1.$ We also get that $H^{i}(T_{2})=0$
unless $i=0,1$ and $H^{i}(F_{2})=0$ unless $i=1,2.$ Therefore, we have
$T_{2}=F_{2}=0.$

We will now proceed to prove that
$\operatorname{Coh}_{1}^{r}(C)\subseteq\mathcal{A}^{{}^{\prime\prime}}.$ We
show that $\mathcal{T}_{1}=\mathcal{T}_{\theta}$ and
$\mathcal{F}_{1}=\mathcal{F}_{\theta}.$ It is easy to show that
$\mathcal{T}_{\theta}\subseteq\mathcal{T}_{1}$ and
$\mathcal{F}_{\theta}\subseteq\mathcal{F}_{1},$ where
$\operatorname{Coh}(C)=(\mathcal{T}_{\theta},\mathcal{F}_{\theta})$ is the
torsion pair described in Remark 2.52 (up to shift) and $\theta\neq 0.$

It is enough to show this for slope stable torsion free sheaves
$E\in\operatorname{Coh}(C)$, such that $E[n]\in\operatorname{Coh}(C)[n].$ Let
$0\rightarrow T\rightarrow E[n]\rightarrow F\rightarrow 0$ be the triangle
induced by the torsion pair $(\mathcal{T}_{1},\mathcal{F}_{1}).$ If $T$ and
$F$ are non-trivial, then as $i_{*}(F[1])\in\mathcal{A}^{{}^{\prime\prime}}$
and $i_{*}(T)\in\mathcal{A}^{{}^{\prime\prime}},$ we get that
$\Im(Z)(i_{*}(T))>0,$ which implies $\mu(T[-n])>-\cot(\theta\pi)$ and
$\Im(Z)(i_{*}(F))\leq 0,$ which implies $\mu(F[-n])\leq-\cot(\theta\pi).$
Since $E$ is $\mu$-stable, we obtain a contradiction.

If $\theta=0,$ then $\mathcal{T}^{\theta}_{1}=\operatorname{Coh}(C)[n]$ and
$\mathcal{F}^{\theta}_{1}=0.$ Since $r\in\mathbb{Z},$ we obtain

$Z_{1}(r,d)=A_{1}d_{1}+Br_{1}+(Cr_{1})i,$

i.e. $D_{1}=0.$ This implies that
$\mathcal{T}_{1}=\operatorname{Coh}_{1}(C)[n].$ Therefore
$\operatorname{Coh}^{r}_{1}(C)\subseteq\mathcal{A}^{{}^{\prime\prime}}.$

######  Remark 4.25.

Let us consider $\sigma\in\Theta_{12}$ and
$g^{\prime}=(T^{\prime},f^{\prime})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}).$
By Proposition 4.21, there is
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ such that for
$\sigma g=(Z,\mathcal{A})$ we can find stability conditions

$\sigma_{1}=(Z_{1},\operatorname{Coh}^{r}(C))\textnormal{ and
}\sigma_{2}=(Z_{\mu},\operatorname{Coh}(C))\in\operatorname{Stab}(C)$

with $\operatorname{Coh}^{r}_{1}(C)\subseteq\mathcal{A}$ and $r>-1,$
$\operatorname{Coh}_{2}(C)\subseteq\mathcal{A}.$ Note that the proof of the
inclusion of the hearts in $\mathcal{A}$ depends only on the stability of
$i_{*}(\mathbb{C}(x))$ and $j_{*}(\mathbb{C}(x))$ for all $x\in C$ and
$i_{*}(\mathcal{L})$ and $j_{*}(\mathcal{L}),$ for every line bundle
$\mathcal{L}.$ As a consequence, we obtain that $\sigma
gg^{\prime}=(Z^{\prime},\mathcal{A}^{\prime})$ satisfies that

$\operatorname{Coh}_{1}^{f\circ f^{\prime}(0)}(C)\subseteq
A^{\prime}\textnormal{ and
}\operatorname{Coh}^{f^{\prime}(0)}_{2}(C)\subseteq\mathcal{A}^{\prime}.$

The following lemma gives us some CP-glued stability conditions in
$\Theta_{12}.$

######  Lemma 4.26.

Let $\sigma=(Z,\mathcal{A})\in\Theta_{12},$ such that there are stability
conditions

$\sigma_{1}=(Z_{1},\operatorname{Coh}^{r}(C))=(T_{1},f_{1})\textnormal{ and
}\sigma_{2}=(Z_{\mu},\operatorname{Coh}(C))\in\operatorname{Stab}(C)$

with $\operatorname{Coh}^{r}_{1}(C)\subseteq\mathcal{A},$
$\operatorname{Coh}_{2}(C)\subseteq\mathcal{A},$ and
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{1}=Z_{1}$ and
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{2}=Z_{\mu}.$ If
$f_{1}(0)=r\geq 0,$ then
$\sigma=\operatorname{gl}_{12}(\sigma_{1},\sigma_{\mu}).$

######  Proof 18.

Since $\sigma_{1}$ and $\sigma_{2}$ satisfy gluing conditions i.e
$f_{1}(0)\geq f_{2}(0)=0,$ then there is a pre-stability condition
$\sigma_{12}=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})$ on
$\mathcal{T}_{C}.$ It follows directly from Proposition 3.36 that
$\sigma=\sigma_{12}.$

We will now study pre-stability conditions
$\sigma=(Z,\mathcal{A})\in\Theta_{12}.$ For

$\sigma_{1}=(Z_{1},\operatorname{Coh}^{r}(C))\textnormal{ and
}\sigma_{2}=(Z_{\mu},\operatorname{Coh}(C))\in\operatorname{Stab}(C)$

we have $\operatorname{Coh}^{r}_{1}(C)\subseteq\mathcal{A},$
$\operatorname{Coh}_{2}(C)\subseteq\mathcal{A},$
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{1}=Z_{1},$
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{2}=Z_{\mu},$ and
$-1<r<0,$ where $r=n+\theta=f_{1}(0)$ and $n\in\mathbb{Z}$ with
$\theta\in[0,1).$ Here $\sigma_{1}$ is given by $(T_{1},f_{1})$ under the
correspondence with $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}).$

######  Remark 4.27.

$l_{*}(\mathbb{C}(x))$ is $\sigma$-stable. Assume that $l_{*}(\mathbb{C}(x))$
is not stable. By Remark 4.9 this implies that $n+\theta=f_{1}(0)\geq 0,$
which contradicts our assumption.

###  4.2. Pre-stability conditions in $\Theta_{23}$ and $\Theta_{31}$

We have the analogous statement of Proposition 4.21 for $\Theta_{23}$ and
$\Theta_{31}.$ We leave the proof to the reader, since it follows exactly the
same steps as the result for $\Theta_{12}.$

######  Proposition 4.28.

For every $\sigma\in\Theta_{23},$ there is a
$g\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ such that for $\sigma
g=(Z,\mathcal{A})$ we can find stability conditions

$\sigma_{3}=(Z_{3},\operatorname{Coh}^{r}(C))\textnormal{ and
}\sigma_{2}=(Z_{\mu},\operatorname{Coh}(C))\in\operatorname{Stab}(C)$

with $\operatorname{Coh}^{r}_{3}(C)\subseteq\mathcal{A},$
$\operatorname{Coh}_{2}(C)\subseteq\mathcal{A},$ $r<0,$
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{3}=Z_{3}$ and
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{2}=Z_{\mu}.$

######  Proposition 4.29.

For every $\sigma\in\Theta_{31},$ there is a
$g\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ such that for $\sigma
g=(Z,\mathcal{A})$ we can find stability conditions

$\sigma_{3}=(Z_{3},\operatorname{Coh}^{r_{3}}(C))\textnormal{ and
}\sigma_{1}=(Z_{1},\operatorname{Coh}^{r_{1}}(C))\in\operatorname{Stab}(C)$

with $\operatorname{Coh}^{r_{3}}_{3}(C)\subseteq\mathcal{A},$
$\operatorname{Coh}^{r_{1}}_{1}(C)\subseteq\mathcal{A},$ with $r_{3}-r_{1}>0,$
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{3}=Z_{3},$
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{D}}_{1}=Z_{1}$ and
$M_{3}-M_{1}=I.$

###  4.3. Non-gluing pre-stability conditions

We will now study pre-stability conditions $\sigma$ on $\mathcal{T}_{C}$
satisfying that $i_{*}(\mathbb{C}(x))$ and $l_{*}(\mathbb{C}(x))$ are stable,
where $j_{*}(\mathbb{C}(x))$ is stable with phase one and
$j_{*}(\mathcal{O}_{C})$ has phase 1/2. Note that as proved in Lemma 4.18, we
have $l_{*}(\mathbb{C}(x))[-1]\in\mathcal{A}.$ We use Lemma 4.18 to prove that
these stability conditions are precisely given by the pair
$(Z_{r},\mathcal{A}_{r})$ constructed in Lemma 3.48.

######  Lemma 4.30.

Let $\sigma=(Z,\mathcal{A})$ be a pre-stability condition on $\mathcal{T}_{C}$
such that $i_{*}(\mathbb{C}(x)),l_{*}(\mathbb{C}(x))$ are $\sigma$-stable and
the object $j_{*}(\mathbb{C}(x))$ and $j_{*}(\mathcal{O}_{C})$ are in
$\mathcal{A}$ and are also $\sigma$-stable with

$Z([j_{*}(\mathbb{C}(x))])=-1\textnormal{ and }Z([j_{*}(\mathcal{O}_{C})])=i.$

Then $\sigma$ is given by the pairs constructed in Lemma 3.48.

######  Proof 19.

We prove the statement for $\sigma\in\Theta_{12},$ the other two cases follow
analogously. By Proposition 4.21, there are stability conditions
$\sigma_{1}=(Z_{1},\operatorname{Coh}^{r}(C))\textnormal{ and
}\sigma_{2}=(Z_{\mu},\operatorname{Coh}(C))\in\operatorname{Stab}(C),$ such
that $\operatorname{Coh}^{r}_{1}(C)\subseteq\mathcal{A}$ and
$\operatorname{Coh}_{2}(C)\subseteq\mathcal{A},$ with
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{1}=Z_{1}$ and
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{2}=Z_{\mu}.$
Note that $-1<f_{1}(0)<0,$ where
$\sigma_{1}=(T_{1},f_{1})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}).$
First of all we show that $Z$ is given by equation 3.19. Our stability
function $Z$ is completely determined by $Z_{1}$ and $Z_{\mu},$ therefore it
has the following form

$Z(r_{1},d_{1},r_{2},d_{2})=Ad_{1}+Br_{1}-d_{2}+i(Cr_{1}+Dd_{1}+r_{2}).$

Let $M\coloneqq T_{1}^{-1}=\begin{bmatrix}-A&B\\\ -D&C\\\ \end{bmatrix}.$
Since $\sigma_{1}$ is a stability condition, we have that $\det(M)>0.$ As
$-1<f_{1}(0)<0,$ then by Lemma 4.18, we obtain $1<\phi_{0}<2$ and $D<0.$

We will now show that $\det(M+I)>0.$ If $\operatorname{Tr}(M)\geq 0,$ there is
nothing to prove because $\det(M+I)=\det(M)+\operatorname{Tr}(M)+1>0.$ Note
that if $\operatorname{Tr}(M)<0,$ then $l_{*}(\mathcal{O}_{C})$ is
$\sigma$-stable. Since $l_{*}(\mathbb{C}(x))$ and $l_{*}(\mathcal{O}_{C})$ are
stable, we obtain $\phi_{5}<\phi_{4}<\phi_{5}+1$ which is equivalent to the
fact that $\det(M+I)>0.$

We consider the torsion pair
$(\mathcal{T},\mathcal{F})=\operatorname{TCoh}(C)$ given in Lemma 4.18. We are
going to show that it is equal to the torsion pair
$(\mathcal{T}^{{}^{\prime}},\mathcal{F}^{{}^{\prime}})$ given by Lemma 3.43.
It is enough to show that $\mathcal{T}^{{}^{\prime}}\subseteq\mathcal{T}$ and
$\mathcal{F}^{{}^{\prime}}\subseteq\mathcal{F}.$

Let us take a torsion-free $\lambda$-semistable object $E=E_{1}\rightarrow
E_{2}\in\mathcal{T}^{\prime},$ which by definition satisfies $\lambda(E)>3/4.$
There is a short exact sequence $0\rightarrow T\rightarrow E\rightarrow
F\rightarrow 0,$ with $T\in\mathcal{T}$ and $F=F_{1}\rightarrow
F_{2}\in\mathcal{F}.$ By Lemma 4.18 and Lemma 3.43 we have that
$i^{!}(E),i^{!}(T)\in\operatorname{Coh}(C).$ It follows that
$i^{!}(F)\in\operatorname{Coh}(C).$

As $F_{1}$ is torsion-free and $F_{1}\twoheadrightarrow F_{2},$ then $F\neq
0\rightarrow G_{2},$ where $G_{2}$ is a torsion sheaf. We apply Remark 3.40
and we obtain that $3/4<\lambda(E)\leq\lambda(F)\leq 3/4,$ which gives us a
contradiction and this implies that $E\in\mathcal{T}.$

Let us take a $\lambda$-semistable torsion-free object $E=E_{1}\rightarrow
E_{2}\in\mathcal{F}^{\prime},$ and by definition holds that $\lambda(E)\leq
3/4.$ We start with the case $\lambda(E)<3/4.$ There is a short exact sequence
given by $0\rightarrow T\rightarrow E\rightarrow F\rightarrow 0,$ with
$T\in\mathcal{T}$ and $F\in\mathcal{F}.$ We get $\lambda(T)\geq 3/4$ and
$\lambda(F)\leq 3/4.$ If $T\neq 0,$ then by $\lambda$-semistability
$3/4\leq\lambda(T)\leq\lambda(E)<3/4,$ which gives us a contradiction and this
implies that $E\in F.$

Let us take a torsion-free $\lambda$-semistable object
$E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{F}^{\prime}$ with
$\lambda(E)=3/4.$ Consider the short exact sequence $0\rightarrow T\rightarrow
E\rightarrow F\rightarrow 0,$ with $T\in\mathcal{T}$ and $F\in\mathcal{F}.$
Note that the inequality $3/4\leq\lambda(T)\leq\lambda(E)=3/4$ holds, so that
$3/4=\lambda(T).$ If $T=T_{1}\xrightarrow{\varphi^{\prime}}T_{2},$ we get
$T[-1]\in\mathcal{P}^{\prime}(1).$ Moreover by Lemma 4.18, we have that
$\mathcal{P}^{\prime}(1)\subseteq\operatorname{TCoh}(C)$ which implies
directly that $T=0.$

Note that for all torsion-free objects $E\in\mathcal{T}^{{}^{\prime}},$ after
using its Harder–Narasimhan filtration, we obtain that $E\in\mathcal{T}.$
Analogously for $E\in\mathcal{F}^{{}^{\prime}}.$ We easily extend the result
above to any object in $\mathcal{T}$ and $\mathcal{F}.$ Consequently
$\mathcal{T}=\mathcal{T}^{{}^{\prime}}$ and
$\mathcal{F}=\mathcal{F}^{{}^{\prime}}.$

######  Remark 4.31.

Under the assumption of the last proposition, for $\sigma\in\Theta_{12}$ we
can define a third stability condition
$\sigma_{3}=((M+I)^{-1},f_{3})\in\operatorname{Stab}(C),$ that comes from the
data given by $l_{*}(\mathbb{C}(x))$ and $l_{*}(\mathcal{O}_{C}),$ where the
integer part of $f_{3}(0)=r_{3}$ is $-1.$ Note that by Lemma 3.50, we obtain
$\operatorname{Coh}^{r_{3}}_{3}(C)\subseteq\mathcal{A}.$

######  Proposition 4.32.

Let $\sigma$ be a pre-stability condition in $\Theta_{12}.$ There is an
element $g\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ such that $\sigma
g$ is given by a CP-glued pre-stability condition or one constructed by
tilting in Subsection 3.2.2.

######  Proof 20.

After applying Proposition 4.21, this follows directly from Lemma 4.26 and
Lemma 4.30.

###  4.4. Characterising CP-glued pre-stability conditions in $\Theta_{12}$

Let us consider the sets $\Theta_{i}$ consisiting of pre-stability conditions,
for $i=1,2$ or $3,$ which are, up to the action of
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ CP-glued with respect to the
semiorthognal decomposition
$\langle\mathcal{C}_{i},{{}^{\perp}}\mathcal{C}_{i}\rangle.$ Note that by
Proposition 3.36, it follows that the set $\Theta_{1}\subseteq\Theta_{12}.$

The aim of this section is to characterise the sets $\Theta_{i}$ inside
$\Theta_{12}.$

Condition (*): If $\sigma\in\Theta_{12}$ satisfies that there are stability
conditions $\sigma_{1}=(\operatorname{Coh}^{r}_{1}(C),Z_{1})$ and
$\sigma_{2}=(\operatorname{Coh}_{2}(C),Z_{\mu})\in\operatorname{Stab}(C)$ with
$\operatorname{Coh}^{r}_{1}(C)\subseteq\mathcal{A},$
$\operatorname{Coh}(C)_{2}\subseteq\mathcal{A},$ and
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{1}=Z_{1}$ and
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{2}=Z_{\mu}.$ Let
$f_{1}(0)=n+\theta,$ with $n\in\mathbb{Z}$ and $\theta\in[0,1),$ we say that
it satisfies condition (*). Let $\sigma\in\Theta_{12}.$ By Proposition 4.21,
we have that there is $g\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$
such that $\sigma g$ satisfies condition (*).

Under condition (*), we define $f_{12}(\sigma)(x)=f_{1}(x)-x.$

From now on we assume that $\sigma$ satisfies condition (*) and
$\sigma_{1}=(T,f_{1}).$ If $f_{12}(\sigma)(0)\geq 0$ by Proposition 4.26, we
already know that $\sigma\in\Theta_{1}.$ Now we assume $f_{12}(\sigma)(0)<0.$
We would like to understand when $\sigma\in\Theta_{12}$ is in $\Theta_{i},$
for $i=1,2,3.$

Recall that by Remark 4.31 there is also
$\sigma_{3}=(Z_{3},\operatorname{Coh}^{r_{3}}(C))$ with
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{3}=Z_{3}$ and
$\operatorname{Coh}^{r_{3}}_{3}(C)\subseteq\mathcal{A},$ and we can also
define $f_{23}(\sigma)(x)=x-f_{3}(x)\textnormal{ and
}f_{31}(\sigma)(x)=f_{3}(x)-f_{1}(x).$

######  Lemma 4.33.

Let $\sigma$ be as in (*), such that $f_{12}(\sigma)(0)<0.$ There is a
$t\in\mathbb{R}$ such that $f_{12}(\sigma)(t)=0$ if and only if
$\sigma\in\Theta_{1}.$

######  Proof 21.

Assume that there is $t\in\mathbb{R}$ such that $f_{12}(\sigma)(t)=0.$ Since
$f_{2}(t)=t,$ this implies that $f_{1}(t)=t.$ Let
$g=(K_{t\pi},f_{t\pi})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}).$ Since
$f_{1}\circ f_{t\pi}(0)=f_{1}(t)=t$ and $f_{2}\circ f_{t\pi}(0)=f_{2}(t)=t,$
as a consequence $f_{12}(\sigma g)(0)=0$ and it follows directly from Lemma
3.28 that $\sigma g=\operatorname{gl}_{12}(\sigma_{1}g,\sigma_{2}g).$

We now assume that $\sigma\in\Theta_{1},$ then there is a
$g\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ such that $\sigma g$
satisfies the gluing conditions for $\sigma_{1}g$ and $\sigma_{2}g$. Without
losing generality, we take
$g=(K_{l\pi},f_{l\pi})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ with
$l\in\mathbb{R}$ Thus, $f_{1}(l)=(f_{1}\circ f_{l\pi})(0)\geq(f_{2}\circ
f_{l\pi})(0)=f_{l\pi}(0)=l.$ By hypothesis $f_{12}(\sigma)(0)<0,$ but also
$f_{12}(\sigma)(l)\geq 0.$ Since $f_{12}(\sigma)(x)$ is a continuous function,
there is a $t\in\mathbb{R}$ that satisfies $f_{12}(\sigma)(t)=0,$ as we
desired.

######  Remark 4.34.

[Fixed point] Since in our case $f_{12}(\sigma)(x)=f_{1}(x)-x,$ we obtain that
$f_{12}(\sigma)(x)=0$ if and only if there is $x\in\mathbb{R}$ such that
$f_{1}(x)=x.$ After using that the restrictions of $f_{1}$ and $T$ to $S^{1}$
agree, the search of points $x\in\mathbb{R}$ such that $f_{1}(x)=x$ reduces to
the study of the eigenvectors of $T.$

Let $M\coloneqq T^{-1}=\begin{bmatrix}-A&B\\\ -D&C\\\ \end{bmatrix}.$

We will now study the characteristic polynomial
$p(x)=x^{2}-\operatorname{Tr}(M)x+\det(M)$ of $M.$ It plays an important role
in determining if $\sigma\in\Theta_{i}$ for some $i=1,2,3.$ The sign of the
discriminant $\Delta(M)=\operatorname{Tr}(M)^{2}-4\det(M)$ of $p(x)$ tells us
about the existence of real eigenvalues.

######  Proposition 4.35.

If $f_{12}(\sigma)(0)<1,$ the discriminant of
$\Delta(M)=\operatorname{Tr}(M)^{2}-4\det(M)$ is non-negative and the
eigenvalues are positive, then there is
$g\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ such that $\sigma
g=(Z,\operatorname{TCoh}^{l}(C)),$ with $l\in\mathbb{R}$ and $-1<l\leq 1.$
Moreover $\sigma\in\Theta_{1}.$

######  Proof 22.

Since $\Delta(M)\geq 0,$ it guarantees the existence of real eigenvalues. By
hypothesis the eigenvalues are positive. The same follows for $T_{1}.$ Let
$\lambda\in\mathbb{R}$ be an eigenvalue of $T_{1}$ and $v\in\mathbb{R}^{2}$
its corresponding eigenvector. We obtain $T_{1}v=\lambda v.$ Let us consider
the polar coordinates of $v=(m\cos(\phi),m\sin(\phi))$ with
$\phi\in(-\pi,\pi]$ and $m\in\mathbb{R}_{>0}.$

We claim now that $\sigma g\in\Theta_{1},$ where
$g=(K_{\phi},f_{\phi})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}).$ First
of all we consider $\sigma_{1}g=(T_{1}K_{\phi},f_{1}\circ f_{\phi}).$ By the
correspondence between $f_{1}\circ f_{\phi}$ and $T_{1}K_{\phi}$ over $S^{1}$
and the fact that $v$ is an eigenvector, we obtain that $(f_{1}\circ
f_{\phi})(0)=\phi/\pi.$ Now we consider $\sigma_{2}g=(K_{\phi},f_{\phi}).$ As
a consequence, $f_{12}(\sigma g)=0.$ By Lemma 3.36, it is clear that $\sigma
g=(Z,\operatorname{TCoh}^{\phi}(C))=\operatorname{gl}_{12}(\sigma_{1}g,\sigma_{2}g).$

######  Remark 4.36.

Note that from Lemma 4.35 and Remark 4.34, we obtain that if
$\sigma\in\Theta_{12}$ satisfies (*), then $\sigma\in\Theta_{1}$ if and only
if the eigenvalues of $M$ are positive. Moreover, by Lemma 4.33, if
$\Delta(M)<0,$ then $\sigma$ can never be in $\Theta_{1}.$

Under the assumption that the discriminant $\Delta(M)$ of $M$ is non-negative
and that the eigenvalues $\lambda_{1},\lambda_{2}$ of $M$ are both negative we
prove the following lemmas.

######  Remark 4.37.

Under the assumption that the discriminant $\Delta(M)$ of $M$ is non-negative
and that the eigenvalues are negative, then if $\det(M+I)>0,$ there are just
two options, either $\lambda_{1},\lambda_{2}<-1\textnormal{ or
}\lambda_{1},\lambda_{2}>-1.$

The pre-stability conditions in $\Theta_{2}$ or $\Theta_{3}$ satisfy that
$l_{*}(\mathcal{L})$ is stable for all line bundles
$\mathcal{L}\in\operatorname{Coh}(C).$ Hence, in order to determine which
$\sigma\in\Theta_{12}$ are in $\Theta_{2}$ or $\Theta_{3}$ we need to study
the stability of $l_{*}(\mathcal{L}).$

######  Lemma 4.38.

Let $\sigma=(Z,\mathcal{A})$ be a pre-stability condition as in (*). Let
$\mathcal{L}\in\operatorname{Coh}(C)$ be a line bundle with
$\deg(\mathcal{L})=d\leq\frac{-C}{D}$. The object $l_{*}(\mathcal{L})$ is
stable in $\sigma$ if and only if the inequality $-Dd^{2}-(A+C)d-B>0$ holds.

######  Proof 23.

Since $\deg(\mathcal{L})=d\leq\frac{-C}{D}$ we have
$i_{*}(\mathcal{L})\in\mathcal{A}.$ By considering the triangle

$\textstyle{j_{*}(\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{l_{*}(\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{*}(\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{*}(\mathcal{L})[1]}$

and the correspondence between slope and phase the result follows.

######  Remark 4.39.

  1. (1)

The discriminant of the quadratic equation $Dd^{2}+(A+C)d+B$ is given by
$(A+C)^{2}-4BD=\Delta(M),$ i.e. it has the same discriminant as $p(x).$

  2. (2)

By Remark 4.10, if $\mathcal{L}$ is a line bundle and
$i_{*}(\mathcal{L})[-1]\in\mathcal{A},$ then $l_{*}(\mathcal{L})$ is stable.

######  Lemma 4.40.

If $\lambda_{1},\lambda_{2}<0,$ then inequality $-Dd^{2}-(A+C)d-B>0$ holds for
all $d\leq-\cot(\theta\pi).$

######  Proof 24.

Let us consider the polynomial $q(x)=Dx^{2}+(A+C)x+B.$ As the discriminant of
$q(x)$ is also $\Delta(M)\geq 0,$ we get that $q(x)$ has real roots
$\mu_{1},\mu_{2}\in\mathbb{R}.$ We assume that $\mu_{1}\leq\mu_{2}.$ It is
enough to conduct the proof for $\mu_{1}\geq\frac{-C}{D}.$ Indeed, if
$\frac{-C}{D}\leq\mu_{1}$ and $d<\frac{-C}{D}$ then $q(x)<0.$ As
$0>2\lambda_{1}=(-A+C)-\sqrt{\Delta(M)},$ we obtain

$\displaystyle\frac{C}{-D}$ $\displaystyle<$
$\displaystyle\frac{-(A+C)-\sqrt{\Delta(M)}}{2D}=\mu_{1},$

as we wanted to prove.

######  Corollary 4.41.

If $\lambda_{1},\lambda_{2}<0,$ then $l_{*}(\mathcal{L})$ is $\sigma$-stable
for all line bundles $\mathcal{L}.$ Moreover, $\sigma$ is in $\Theta_{23}$ and
$\Theta_{31}.$

######  Proof 25.

This follows directly from Lemma 4.38, Lemma 4.39 and Lemma 4.40.

######  Lemma 4.42.

  1. (1)

There is $t\in\mathbb{R}$ such that $f_{31}(\sigma)(t)=1$ if and only if
$\sigma\in\Theta_{3}.$

  2. (2)

There is a $t\in\mathbb{R}$ such that $f_{23}(\sigma)(t)=1$ if and only if
$\sigma\in\Theta_{2}.$

######  Proof 26.

The proof goes along the lines of Lemma 4.33.

######  Proposition 4.43.

  1. (1)

If $\lambda_{1},\lambda_{2}<-1,$ then $\sigma\in\Theta_{3}.$

  2. (2)

If $0>\lambda_{1},\lambda_{2}>-1,$ then $\sigma\in\Theta_{2}.$

######  Proof 27.

The proof goes exactly as the one of Lemma 4.35.

######  Remark 4.44.

  1. (1)

Note that from Proposition 4.43 and Lemma 4.42, we obtain that if
$\sigma\in\Theta_{12}$ satisfies (*), then $\sigma\in\Theta_{3}$ if and only
if the eigenvalues $\lambda_{1},\lambda_{2}$ of $M$ are $<-1.$

  2. (2)

Note that from Proposition 4.43 and Lemma 4.42, we obtain that if
$\sigma\in\Theta_{12}$ satisfies (*), then $\sigma\in\Theta_{2}$ if and only
if the eigenvalues $\lambda_{1},\lambda_{2}$ of $M$ satisfy
$0>\lambda_{1},\lambda_{2}>-1.$

######  Remark 4.45.

Note that if $\sigma\in\Theta_{12}$ satisfies (*), then either
$\sigma\in\Theta_{1}$ or $\sigma\in\Theta_{12}\cap\Theta_{23}.$ Indeed, if the
eigenvalues of $M$ are positive, we have that $\sigma\in\Theta_{1}.$ If the
eigenvalues are negative, this follows from Corollary 4.41. If $\Delta(M)<0,$
it follows from Lemma 4.38. This remark is useful when studying
$\Theta_{23}$(or $\Theta_{31}$) because either $\sigma\in\Theta_{2}$ or
$\sigma\in\Theta_{12}\cap\Theta_{23},$ thus we can use our classification
inside $\Theta_{12}$ to understand $\Theta_{23}.$

######  Remark 4.46.

Note that if $\sigma\in\Theta_{23},$ then by Proposition 4.28, up to the
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action we can find stability
conditions
$\sigma_{3}=(Z_{3},\operatorname{Coh}^{r}(C))=(T_{3},f_{3})\textnormal{ and
}\sigma_{2}=(Z_{\mu},\operatorname{Coh}(C))\in\operatorname{Stab}(C)$ with
$\operatorname{Coh}^{r}_{3}(C)\subseteq\mathcal{A},$
$\operatorname{Coh}_{2}(C)\subseteq\mathcal{A},$ $r<0,$
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{3}=Z_{3}$ and
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{2}=Z_{\mu}.$ If
$i_{*}(\mathbb{C}(x))$ is $\sigma$-stable for $x\in C$ and $\Delta(M_{3})<0,$
with $M_{3}\coloneqq T_{3}^{-1},$ by following the steps of Lemma 4.38 we can
show that $i_{*}(\mathcal{L})$ is $\sigma$-stable for all line bundles
$\mathcal{L}.$ As a consequence, by Lemma 4.30, we have that $\sigma$ is
given, up to the $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action, by a
pre-stability condition constructed in Lemma 3.48. Moreover, $M_{3}=I+M,$ the
$M$ appearing in Lemma 3.48 and $\Delta(M)=\Delta(M_{3})<0.$ Analogously for
the case $\sigma\in\Theta_{31}.$

######  Theorem 4.47.

For all pre-stability conditions $\sigma$ on $\mathcal{T}_{C},$ we have that

$\sigma\in\Theta_{1}\cup\Theta_{2}\cup\Theta_{3}\cup\Gamma,$

where $\Gamma$ is the set of pre-stability conditions, which up to the
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action is given by Lemma 3.48
with $\Delta(M)<0.$ Moreover, note that $\Gamma\subseteq\Theta_{ij},$ where
$ij=12,23,31.$

######  Proof 28.

By Theorem 4.13, we have that
$\sigma\in\Theta_{12}\cup\Theta_{23}\cup\Theta_{31}.$ By Serre duality, it
suffices to check for $\sigma\in\Theta_{12}.$ By Proposition 4.21, we have
that up to the action of $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$
there are $\sigma_{1}=(\operatorname{Coh}^{r}_{1}(C),Z_{1})=(T,f)$ and
$\sigma_{2}=(\operatorname{Coh}_{2}(C),Z_{\mu})\in\operatorname{Stab}(C)$ with
$\operatorname{Coh}^{r}_{1}(C)\subseteq\mathcal{A},$
$\operatorname{Coh}(C)_{2}\subseteq\mathcal{A},$ and
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{1}=Z_{1},$
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}}}_{2}=Z_{\mu}$ with
$f(0)\geq-1.$ We reduce to the study of these pre-stability conditions,
because the sets $\Theta_{i}$ and $\Gamma,$ for $i=1,2,3,$ are defined up to
the $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action. If $f(0)\geq 0,$
then by Lemma 4.26 $\sigma\in\Theta_{1}.$ If $-1<f(0)<0,$ we classify these
pre-stability conditions in two sets if $\Delta(M)\geq 0$ or $\Delta(M)<0.$ If
$\Delta(M)<0$ then $\sigma\in\Gamma.$ If $\Delta(M)\geq 0,$ with positive
eigenvalues then by Lemma 4.35 we get $\sigma\in\Theta_{1}.$ If the
eigenvalues are smaller than $-1,$ then by Lemma 4.43 we have that
$\sigma\in\Theta_{3}$ and if the eigenvalues are between $0$ and $-1$ by Lemma
4.43, we obtain that $\sigma\in\Theta_{2}$.

######  Remark 4.48.

Note that by Lemma 4.33 and Lemma 4.42 we have that

$\Theta_{i}\cap\Theta_{j}=\emptyset\textnormal{ and
}\Theta_{i}\cap\Gamma=\emptyset\textnormal{ for }i,j\in\\{1,2,3\\}.$

##  5\. Support property for $\mathcal{T}_{C}$, with $g(C)\geq 1$

In this section, we prove the support property for the already constructed
pre-stability conditions. We start by studying CP-glued pre-stability
conditions $\sigma=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})$ on
$\mathcal{T}_{C}=\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle,$ with
$\sigma_{i}=(Z_{i},\mathcal{A}_{i})\in\operatorname{Stab}(D^{b}(C)),$ for
$i=1,2.$ By Theorem 2.49 there is always
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ such that
$\sigma_{1}=\sigma_{2}g.$ Note that the gluing condition states precisely that
$f(0)\geq 0$. Therefore, the proof of the support property falls naturally
into the cases $f(0)\geq 1$ and $1>f(0)\geq 0.$ More precisely we reduce the
proof of the support property to three cases. Namely, in Lemma 5.1 we prove it
for pre-stability conditions satisfying that $f(0)\geq 1,$ in Lemma 5.20 we
prove it for CP-glued pre-stability condition with $1>f(0)\geq 0$ and
$\Delta(T^{-1})>0.$ In Lemma 5.22 for CP-glued pre-stability conditions with
$1>f(0)\geq 0$ and negative discriminant. Finally, in Subsection 5.3 we prove
it for $\sigma\in\Gamma.$ We give some inequalities to study the
$\sigma$-semistable objects in our pre-stability conditions. These
inequalities follow closely the steps of [8, Section 3]. In Subsection 5.3 we
prove the support property under the assumption that $g(C)=1,$ for non-gluing
pre-stability conditions with $\Delta(M)<0.$ We conjecture that for $g(C)>1$
the same result holds.

###  5.1. Support property for CP-glued pre-stability conditions

We start by proving the support property for pre-stability conditions with a
stronger orthogonality condition.

######  Lemma 5.1.

If the pair
$\sigma=(Z,\mathcal{A})=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})$ is a
CP-glued pre-stability condition satisfying that
$\operatorname{Hom}_{\mathcal{T}_{C}}^{\leq
1}(i_{*}\mathcal{A}_{1},j_{*}\mathcal{A}_{2})=0$ or equivalently $f(0)\geq 1,$
then it satisfies the support property and in consequence it is a Bridgeland
stability condition.

######  Proof 29.

We use the following notation $Z_{1}([E])=Z_{1}([i^{*}(E)])$ and
$Z_{2}([E])=Z_{2}([j^{!}(E)]).$

First note that we can linearly extend $Z$ and the homomorphism induced by the
exact functors $i^{*},j^{!}$ to
$\mathcal{N}(\mathcal{T}_{C})\otimes\mathbb{R}\cong\mathbb{R}^{4}.$ We define
the quadratic form

$Q\colon\mathcal{N}(\mathcal{T}_{\mathcal{C}})\otimes\mathbb{R}\rightarrow\mathbb{R}\textnormal{
as }Q(v)=\Im(Z_{1}(v))\Im(Z_{2}(v))+\Re(Z_{1}(v))\Re(Z_{2}(v)),$

where $\Re(\alpha)$ and $\Im(\alpha)$ are the real and the imaginary parts of
$\alpha\in\mathbb{R}^{4}$ respectively.

By the linearity of $Z$, it is clear that $Q$ is a quadratic form. We first
show that it is negative definite on
$\operatorname{Ker}(Z)=\\{v\in\mathbb{R}^{4}\mid\Re(Z_{1}(v))=-\Re(Z_{2}(v))\textnormal{
and }\Im(Z_{1}(v))=-\Im(Z_{2}(v))\\}.$ Indeed, we have that
$Q(v)=-\Im(Z_{1}(v))^{2}-\Re(Z_{1}(v))^{2}\leq 0,$ for
$v\in\operatorname{Ker}(Z).$ Note that $Q(v)=0,$ implies that $Z_{i}(v)=0,$
for $i=1,2.$ Then $v=0$ as $\operatorname{rank}(\mathcal{N}(D^{b}(C))=2.$

Let $E=E_{1}\xrightarrow{\varphi}E_{2}$ be a $\sigma$-semistable object. By
[5, Lemma A.6] it is enough to show that $Q(E)\geq 0$ for $\sigma$-stable
objects. Note that $\varphi=0.$ Indeed, as $\sigma$ is a CP-glued pre-
stability condition, by the definition of
$\operatorname{gl}_{12}(\mathcal{A}_{1},\mathcal{A}_{2}),$ we have that
$E_{1}\in\mathcal{A}_{1}$ and $E_{2}\in\mathcal{A}_{2}.$ As
$\varphi\in\operatorname{Hom}_{D^{b}(C)}(E_{1},E_{2})=\operatorname{Hom}_{\mathcal{T}_{C}}(i_{*}(E_{1}),j_{*}(E_{2})[1])$
and by hypothesis
$\operatorname{Hom}_{\mathcal{T}_{C}}(i_{*}(E_{1}),j_{*}(E_{2})[1])=0,$ we
obtain that $\varphi=0.$

Since $E$ is $\sigma$-stable and by the definition of a CP-glued heart, we
have $i_{*}(E_{1}),j_{*}(E_{2})\in\mathcal{A}.$ It follows from the short
exact sequence $0\rightarrow j_{*}(E_{2})\rightarrow E\rightarrow
i_{*}(E_{1})\rightarrow 0$ $E_{1}=0$ or $E_{2}=0.$ As a consequence,
$Q([E])=0.$

We will now follow the steps of [8, Section 3] in order to study the
$\sigma$-semistable objects in the CP-glued pre-stability conditions
$\sigma=(Z,\mathcal{A})=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2}),$ on
$\mathcal{T}_{C}$ such that $\sigma_{1}=\sigma_{2}g$ where $g=(T,f)$ in
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ with $f(0)=0.$ Note that by
the same argument as in Proposition 4.35, these are precisely, up to the
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action, the pre-stability
conditions with $1>f(0)\geq 0$ and positive discriminant.

We have that $g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ has the
following form $T^{-1}=\begin{bmatrix}-A&B\\\ 0&C\end{bmatrix},$ with $C>0$
and $\det(T)>0.$ This implies that
$\mathcal{A}_{1}=\mathcal{A}_{2}=\mathcal{H}.$ We obtain that
$\sigma_{1}=(Z_{1},\mathcal{H})$ and $\sigma_{2}=(Z_{2},\mathcal{H}).$ By
definition we have

(5.1) $\displaystyle\Re(Z_{1}(w))$ $\displaystyle=$
$\displaystyle-A\Re(Z_{2}(w)))+B\Im(Z_{2}(w)),$ $\displaystyle\Im(Z_{1}(w))$
$\displaystyle=$ $\displaystyle C\Im(Z_{2}(w))\textnormal{ where
}w\in\mathbb{Z}^{4}.$

We use the following notation

(5.2) $\displaystyle d_{2}$ $\displaystyle=$
$\displaystyle-\Re(Z_{2}([j^{!}(E]))\textnormal{, }d_{1}$ $\displaystyle=$
$\displaystyle-\Re(Z_{2}([i^{*}(E)])),$ $\displaystyle r_{2}$ $\displaystyle=$
$\displaystyle\Im(Z_{2}([j^{!}(E)]))\textnormal{ and }r_{1}$ $\displaystyle=$
$\displaystyle\Im(Z_{2}([i^{*}(E)])).$

Therefore, we have $Z_{2}([j^{!}(E)])=-d_{2}+ir_{2}\textnormal{ and
}Z_{1}([i^{*}(E)])=Ad_{1}+Br_{1}+i(Cr_{1}).$ Note that for every
$E\in\mathcal{B},$ we have that $r_{1},r_{2}\geq 0.$ If $r_{1},r_{2}\neq 0,$
we define $\mu_{\sigma}(E)=\frac{-Ad_{1}-Br_{1}+d_{2}}{Cr_{1}+r_{2}}.$

We obtain the following inequalities.

######  Lemma 5.2.

If $E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{A}$ with $r_{1},r_{2}\neq 0$
is a $\sigma$-semistable object and $[\varphi]\neq 0,$ then

$-B\geq(A+C)\mu_{\sigma}(E)$

.

######  Proof 30.

As $\mathcal{H}$ is an abelian category, we can compute
$\operatorname{Ker}(\varphi),\operatorname{Coker}(\varphi)\in\mathcal{H}$ and
by the definition of $\mathcal{A},$ we obtain morphisms $E\twoheadrightarrow
i_{*}(\operatorname{Img}(\varphi))$ and
$j_{*}(\operatorname{Img}(\varphi))\hookrightarrow E$ in $\mathcal{A}.$

Let
$Z_{2}(\operatorname{Img}(\varphi))=-d^{\prime\prime}_{1}+r^{\prime\prime}_{1}i.$
Note that $r^{\prime\prime}_{1}\neq 0.$ Indeed, if $r^{\prime\prime}_{1}=0,$
then $\phi(j_{*}(\operatorname{Img}(\varphi)))=1,$ and by the
$\sigma$-semistability
$\phi(j_{*}(\operatorname{Img}(\varphi)))\leq\phi(E)\leq 1,$ therefore
$\operatorname{Img}(Z(E))=0,$ which implies that $r_{1},r_{2}=0,$ which is a
contradiction.

From these short exact sequences and the correspondence between slope and
phase, we obtain that

$\frac{d^{\prime\prime}_{1}}{r^{\prime\prime}_{1}}\leq\mu_{\sigma}(E)\textnormal{
and
}\mu_{\sigma}(E)\leq\frac{-Ad^{\prime\prime}_{1}-Br^{\prime\prime}_{1}}{Cr^{\prime\prime}_{1}}.$

It follows that
$\mu_{\sigma}(E)\leq\frac{-A}{C}\frac{d^{\prime\prime}_{1}}{r^{\prime\prime}_{1}}-\frac{B}{C},$
then $\mu_{\sigma}(E)\leq\frac{-A}{C}\mu_{\sigma}(E)-\frac{B}{C}.$ As we have
that $-A,C>0,$ therefore we obtain $\mu_{\sigma}(E)(A+C)\leq-B.$

######  Example 5.3.

Consider $\sigma_{\alpha}$ as in Example 3.37 on $\mathcal{T}_{C}.$ As
$\sigma_{\alpha}=\operatorname{gl}_{12}(\sigma_{\mu}g,\sigma_{\mu})$ with
$g=(T,f),$ where $T^{-1}=\begin{bmatrix}1&\alpha\\\ 0&1\end{bmatrix}$ and
$f(0)=0.$ If there are $\sigma_{\alpha}$-semistable objects, as $A+C=0$ and
$-B=\alpha,$ then $\alpha\geq 0$ (see [8, Proposition 3.13]).

######  Lemma 5.4.

If $E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{A}$ with $r_{1},r_{2}\neq 0$
and $[\varphi]\neq 0$ is $\sigma$-semistable, then

$-Ad_{1}+d_{2}-\mu_{\sigma}(E)(r_{2}-Ar_{1})\leq 0.$

######  Proof 31.

We clearly have

$\mu_{\sigma}(E)=\frac{-Ad_{1}+d_{2}-Br_{1}}{Cr_{1}+r_{2}}=\frac{-Ad_{1}+d_{2}}{Cr_{1}+r_{2}}+\frac{-Br_{1}}{Cr_{1}+r_{2}}.$

By Lemma 5.2, we obtain

$\displaystyle\mu_{\sigma}(E)$ $\displaystyle\geq$
$\displaystyle\frac{-Ad_{1}+d_{2}}{Cr_{1}+r_{2}}+(A+C)\frac{\mu_{\sigma}(E)r_{1}}{Cr_{1}+r_{2}},$
$\displaystyle\mu_{\sigma}(E)(r_{2}-Ar_{1})$ $\displaystyle\geq$
$\displaystyle-Ad_{1}+d_{2}.$

######  Lemma 5.5.

If $E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{A}$ with $r_{1},r_{2}\neq 0$
with $[\varphi]\neq 0$ is a $\sigma$-semistable object, then

$(r_{2}-r_{1})\mu_{\sigma}(E)\leq d_{2}-d_{1}$

######  Proof 32.

If $\operatorname{Ker}(\varphi)=0,$ we have the exact sequence $0\rightarrow
l_{*}(E_{1})\rightarrow E\rightarrow
j_{*}(\operatorname{Coker}(\varphi))\rightarrow 0.$ By the
$\sigma$-semistability of $E,$ we directly obtain
$\mu_{\sigma}(E)\leq\frac{d_{2}-d_{1}}{r_{2}-r_{1}}.$

If $\operatorname{Coker}(\varphi)=0,$ we have the exact sequence $0\rightarrow
i_{*}(\operatorname{Ker}(\varphi))\rightarrow E\rightarrow
l_{*}(E_{2})\rightarrow 0.$ We obtain

$\mu_{\sigma}(E)\geq\frac{-A(d_{1}-d_{2})-B(r_{1}-r_{2})}{C(r_{1}-r_{2})}=\frac{-A(d_{1}-d_{2})}{C(r_{1}-r_{2})}+\frac{-B}{C}.$

By Lemma 5.2, we have
$\mu_{\sigma}(E)\geq\mu_{\sigma}(E)+\mu_{\sigma}(E)\frac{A}{C}-\frac{A(d_{1}-d_{2})}{C(r_{1}-r_{2})}.$
Since we assumed $-A,C\geq 0,$ we obtain directly that

$\mu_{\sigma}(E)\geq\frac{d_{1}-d_{2}}{r_{1}-r_{2}}.$

As in this case $r_{1}-r_{2}\geq 0,$ because
$\operatorname{rk}(\operatorname{Ker}(\varphi))=r_{1}-r_{2}$ and
$\operatorname{Ker}(\varphi)\in\mathcal{H},$ which implies

$\mu_{\sigma}(E)(r_{2}-r_{1})\leq(d_{2}-d_{1}).$

Now we assume that $\operatorname{Coker}(\varphi)\neq 0$ and
$\operatorname{Ker}(\varphi)\neq 0.$ From the morphism
$i_{*}(\operatorname{Ker}(\varphi))\hookrightarrow E$ follows that
$\frac{-Ad^{\prime}_{1}-Br^{\prime}_{1}}{Cr^{\prime}_{1}}\leq\mu_{\sigma}(E),$
where $Z_{2}(\operatorname{Ker}(\varphi))=-d^{\prime}_{1}+r^{\prime}_{1}i.$
Note that $r^{\prime}_{1}\neq 0.$ If $r^{\prime}_{1}=0,$ then
$\phi(i_{*}(\operatorname{Ker}(\varphi)))=1$ and by $\sigma$-semistability we
have that $1=\phi(i_{*}(\operatorname{Ker}(\varphi)))\leq\phi(E)\leq 1,$ which
implies that $r_{1},r_{2}=0,$ and this gives us a contradiction.

Since
$\frac{-Ad^{\prime}_{1}}{Cr^{\prime}_{1}}\leq\mu_{\sigma}(E)+\frac{B}{C},$ we
obtain

(5.3)
$-Ad^{\prime}_{1}r_{1}+\mu_{\sigma}(E)r_{2}r^{\prime}_{1}-r^{\prime}_{1}(-Ad_{1}+d_{2})\leq
0.$

Note that if $r_{2}-r^{\prime\prime}_{1}\neq 0,$ then from the morphism
$E\twoheadrightarrow j_{*}(\operatorname{Coker}(\varphi))$ follows that

(5.4)
$\mu_{\sigma}(E)\leq\frac{d_{2}-d^{\prime\prime}_{1}}{r_{2}-r^{\prime\prime}_{1}},\\\
$

where
$Z_{2}(\operatorname{Img}(\phi))=-d^{\prime\prime}_{1}+ir^{\prime\prime}_{1},$
as before.

As $d^{\prime\prime}_{1}=d_{1}-d^{\prime}_{1}$ and
$r^{\prime\prime}_{1}=r_{1}-r^{\prime}_{1},$ after multiplying (5.4) by
$-Ar_{1}$ and adding (5.3) we obtain

$Ar_{1}(d_{2}-d_{1})-r^{\prime}_{1}(-Ad_{1}+d_{2})+\mu_{\sigma}(E)(-Ar_{1}(r_{2}-r_{1})+r^{\prime}_{1}(-Ar_{1}+r_{2}))\leq
0.$

By Lemma 5.4, we have $d_{1}-d_{2}+\mu_{\sigma}(E)(r_{2}-r_{1})\leq 0.$ As a
consequence, we get

$\mu_{\sigma}(E)(r_{2}-r_{1})\leq d_{2}-d_{1}.$

If $r_{2}-r^{\prime\prime}_{1}=0,$ then we obtain that
$d_{2}-d^{\prime\prime}_{1}\geq 0.$ Due to the fact that
$d^{\prime\prime}_{1}=d_{1}-d^{\prime}_{1}$ and
$r^{\prime\prime}_{1}=r_{1}-r^{\prime}_{1},$ we get that

$-Ar_{1}(d_{1}-d^{\prime}_{1}-d_{2})\leq 0.$

Adding the last inequality with (5.3), we have

$\displaystyle\mu_{\sigma}(E)r_{2}r^{\prime}_{1}-r^{\prime}_{1}(-Ad_{1}+d_{2})-Ar_{1}(d_{1}-d_{2})$
$\displaystyle\leq$ $\displaystyle 0.$

By Lemma 5.4, we obtain

$\displaystyle\mu_{\sigma}(E)r_{2}r^{\prime}_{1}-r^{\prime}_{1}(\mu_{\sigma}(E)(r_{2}-Ar_{1})-Ar_{1}(d_{1}-d_{2}))$
$\displaystyle\leq$ $\displaystyle 0.$

As $-A>0,$ we conclude that $\mu_{\sigma}(E)(r_{2}-r_{1})\leq(d_{2}-d_{1}).$

######  Lemma 5.6.

Let $\sigma=(Z,\mathcal{A})$ be as above with
$Z=Ad_{1}+Br_{1}-d_{2}+i(Cr_{1}+r_{2}).$ If there is a $\sigma$-semistable
object with $r_{2}>r_{1}>0$ then $Cy+Ax\leq-B,$ where $x=\frac{d_{1}}{r_{1}}$
and $y=\frac{d_{2}}{r_{2}}.$ Moreover, if $[\varphi]\neq 0$ then $y-x\geq 0,$
and

$-B\in[Cy+Ax,(Ax+Cy+y-x-\frac{r_{1}}{r_{2}}x(A+C))\frac{r_{2}}{r_{2}-r_{1}}].$

######  Proof 33.

Let $E=E_{1}\xrightarrow{\varphi}E_{2}$ be a $\sigma$-semistable object. Let
us consider the following short exact sequence $0\rightarrow
j_{*}(E_{2})\rightarrow E\rightarrow i_{*}(E_{1})\rightarrow 0.$ By the
semistability of $E$ and the correspondence between slope and phase it follows
that

(5.5)
$\frac{d_{2}}{r_{2}}\leq\mu_{\sigma}(E)\leq\frac{-Ad_{1}-Br_{1}}{Cr_{1}}.$

Thus it is implied that $Cr_{1}d_{2}+Ad_{1}r_{2}\leq-\frac{B}{C}r_{1}r_{2},$
as $r_{1},r_{2},C>0.$ We obtain $Cy+Ax\leq-B.$

As $\varphi\neq 0,$ by Lemma 5.5, we have the following inequality
$\frac{-Ad_{1}-Br_{1}+d_{2}}{Cr_{1}+r_{2}}\leq\frac{d_{2}-d_{1}}{r_{2}-r_{1}}.$
From this equation we obtain
$-B\leq\frac{Cd_{2}r_{1}+Ad_{1}r_{2}-(C+A)d_{1}r_{1}+d_{2}r_{1}-d_{1}r_{2}}{(r_{2}-r_{1})r_{1},}.$
After dividing both sides of the inequality into $r_{1}r_{2}$ we get

$-B\in[Cy+Ax,(Ax+Cy+y-x-\frac{r_{1}}{r_{2}}x(A+C))\frac{r_{2}}{r_{2}-r_{1}}].$

From the equation $Cy+Ax\leq-B,$ we obtain

$\displaystyle Cy+Ax$ $\displaystyle\leq$
$\displaystyle(Ax+Cy+y-x-\frac{r_{1}}{r_{2}}x(-A-C))\frac{r_{2}}{r_{2}-r_{1}},$
$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle(y-x)(r_{2}+Cr_{1}).$

As $r_{2}+Cr_{1}>0,$ then we obtain $y-x\geq 0$.

We will now write Lemma 5.6 explicitly in terms of the
$r_{1},r_{2},d_{1},d_{2}$ from the Grothendieck group.

######  Lemma 5.7.

Let
$\sigma=(Z,\operatorname{TCoh}^{\theta}(C))=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})$
be a pre-stability condition on $\mathcal{T}_{C}$ with
$\sigma_{i}=(Z_{i},\operatorname{Coh}^{\theta}(C))$ and $\theta\in[0,1).$ If
$Z_{2}(d_{2},r_{2})=A_{2}d_{2}+Br_{2}+i(C_{2}r_{2}+D_{2}d_{2})$ and
$\sigma_{1}=\sigma_{2}g,$ were
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ and

$T^{-1}=\begin{bmatrix}-A&B\\\ 0&C\end{bmatrix}$

with $C>0$ and there is a $\sigma$-semistable object
$E\in\operatorname{TCoh}^{\theta}(C)$ with $[E]=[r_{1},d_{1},r_{2},d_{2}]$ and
$C_{2}r_{2}+D_{2}d_{2}>C_{2}r_{1}+D_{2}d_{1}>0,$ then $Cy+Ax\leq-B,$ where
$x=\frac{-A_{2}d_{1}-B_{2}r_{1}}{C_{2}r_{1}+D_{2}d_{1}}$ and
$y=\frac{-A_{2}d_{2}-Br_{2}}{C_{2}r_{2}+D_{2}d_{2}}.$ Moreover, if
$[\varphi]\neq 0$ then $y-x\geq 0,$ and

$-B\in[Cy+Ax,(Ax+Cy+y-x-\frac{C_{2}r_{1}+D_{2}d_{1}}{C_{2}r_{2}+D_{2}d_{2}}y(A+C))\frac{C_{2}r_{2}+D_{2}d_{2}}{C_{2}r_{2}+D_{2}d_{2}-C_{2}r_{1}-D_{2}d_{1}}].$

###  5.2. Duality and semistability on $\mathcal{T}_{C}$

The aim of this subsection is to prove an analogous statement to Lemma 5.6 for
$\sigma$-stable objects $E$ with $r_{1}>r_{2}>0,$ where $\sigma$ is a pre-
stability condition on $\mathcal{T}_{C}.$ We follow closely [22] to define the
derived dual in this case and to show that it induces an anti-autoequivalence.

Now we consider the following functor
$\mathbb{D}=R\mathcal{H}om(-,\mathcal{O}_{C})\colon
D^{b}(C)^{\rm{op}}\rightarrow D^{b}(C).$ By [31, Section 2.3], we have that
$\mathbb{D}$ is an equivalence of categories and
$\mathbb{D}^{2}=\operatorname{Id}.$

Let us consider the functor

$\displaystyle\mathcal{H}om(-,\mathcal{O}_{X})_{\operatorname{TCoh}(C)}\colon\operatorname{TCoh}(C)$
$\displaystyle\rightarrow$ $\displaystyle\operatorname{TCoh}(C)$
$\displaystyle E_{1}\xrightarrow{\varphi}E_{2}$ $\displaystyle\mapsto$
$\displaystyle\mathcal{H}om(E_{2},\mathcal{O}_{X})\xrightarrow{\mathcal{H}om(\varphi)}\mathcal{H}om(E_{1},\mathcal{O}_{X}).$

######  Lemma 5.8.

Every bounded object $E$ in $\operatorname{Kom}^{b}(\operatorname{TCoh}(C))$
is quasi-isomorphic to a complex
$F\in\operatorname{Kom}^{b}(\operatorname{TCoh}(C))$ of locally free sheaves
in $\mathcal{T}_{C}.$

######  Proof 34.

It is enough to show that for
$E=E_{1}\xrightarrow{\varphi}E_{2}\in\operatorname{TCoh}(C)$ there is
$F\in\operatorname{TCoh}(C)$ of locally free sheaves with $F\twoheadrightarrow
E.$ There are locally-free sheaves $F_{i}$ and surjective morphisms
$F_{i}\xrightarrow{\pi_{i}}E_{i}$ for $i=1,2.$ Let us consider the triple
$G=F_{1}\rightarrow F_{1}\oplus F_{2}$ as the inclusion of the first
component. Note that $F_{1}\oplus F_{2}$ is also locally free. We have a
surjective morphism $G\rightarrow E$ in $\operatorname{TCoh}(C)$.

For any bounded acyclic object
$E\in\operatorname{Kom}^{b}(\operatorname{TCoh}(C))$ of locally free sheaves,
we have that $\mathcal{H}om(E,\mathcal{O}_{C})_{\operatorname{TCoh}(C)}$ is
acyclic. As a consequence, the class of objects with locally free sheaves as
components in $\operatorname{TCoh}(C)$ is adapted for the left exact functor
$\mathcal{H}om(-,\mathcal{O}_{C})_{\operatorname{TCoh}(C)}$ and by [27, Rem.
2.51], we can define the right derived functor
$R\mathcal{H}om(-,\mathcal{O}_{C})_{\mathcal{T}_{C}}\colon\mathcal{T}_{C}^{\rm{op}}\rightarrow\mathcal{T}_{C}.$

######  Proposition 5.9.

The functor $\mathbb{D}_{1}\coloneqq
R\mathcal{H}om(-,\mathcal{O}_{C})_{\operatorname{TCoh}(C)}\colon\mathcal{T}_{C}^{\rm{op}}\rightarrow\mathcal{T}_{C}$
is an equivalence of categories.

######  Proof 35.

Note that
$\mathcal{T}_{C}^{\rm{op}}=\langle\mathcal{C}_{2}^{\rm{op}},\mathcal{C}_{1}^{\rm{op}}\rangle.$
Moreover, we have that
${\left.\kern-1.2pt\mathbb{D}_{1}\vphantom{\big{|}}\right|_{\mathcal{C}_{2}^{\rm{op}}}}\colon\mathcal{C}_{2}^{\rm{op}}\rightarrow\mathcal{C}_{1}$
and
${\left.\kern-1.2pt\mathbb{D}_{1}\vphantom{\big{|}}\right|_{\mathcal{C}_{1}^{\rm{op}}}}\colon\mathcal{C}_{1}^{\rm{op}}\rightarrow\mathcal{C}_{2}$
are equivalences of categories. As
$\mathbb{D}_{1}(\mathcal{C}_{3}^{\rm{op}})\subseteq C_{3},$ by [30, Lemma 1.3]
we obtain that $\mathbb{D}_{1}$ is an equivalence of categories.

We first study the functor $\mathbb{D}$ on $D^{b}(C),$ where $C$ is a curve
with $g(C)\geq 1.$ By [27, Corollary 3.40], we obtain that
$\mathbb{D}(\mathbb{C}(x))=\mathbb{C}(x)[-1]$ and for a locally free sheaf
$E\in\operatorname{Coh}(C),$ we obtain that
$\mathbb{D}(E)=E^{\vee}=\mathcal{H}om(E,\mathcal{O}_{C})$ wich satisfies that
$\deg(E^{\vee})=-\deg(E)\textnormal{ and
}\operatorname{rank}(E)=\operatorname{rank}(E^{\vee}).$ Moreover, in general
it follows that if $E\in D^{b}(C)$ and $[E]=[r,d]$ then
$[\mathbb{D}(E)]=[r,-d].$

Let us consider the following torsion pair

######  Remark 5.10.

Let $\sigma=(Z,\mathcal{A})\in\operatorname{Stab}(D^{b}(C)).$ We have the
following torsion pair on $\mathcal{A}$.

$\mathcal{T}=\\{E\in\mathcal{A}\mid\phi_{\sigma}(E)=1\\},$
$\mathcal{F}=\\{E\in\mathcal{A}\mid\textnormal{Its
}\sigma\textnormal{-semistable factors }F_{i}\textnormal{ satisfy
}\phi_{\sigma}(F_{i})<1\\}.$

If $\operatorname{Coh}^{\theta}(C)\subseteq\mathcal{T}_{C}$ with
$\theta\in[0,1),$ we rename the torsion pair given in Remark 5.10 because
$\operatorname{Coh}^{\theta}(C)=(T_{\sigma}^{\theta},F_{\sigma}^{\theta}).$
Recall that by Remark 2.52, we also have that
$\operatorname{Coh}^{\theta}(C)=(\mathcal{F}_{\theta}[1],\mathcal{T}_{\theta})$
where $\mathcal{T}_{\theta}=\mathcal{P}_{\mu}(\theta,1]$ and
$\mathcal{F}_{\theta}=\mathcal{P}_{\mu}(0,\theta]$ in $\operatorname{Coh}(C).$

######  Definition 5.11.

Let
$\sigma=(Z(r,d)=Ad+Br+i(Cr+Dd),\operatorname{Coh}^{\theta}(C))=\operatorname{Stab}(C).$
We define $\sigma^{\vee}\coloneqq(Z(r,d)=Ad-Br+i(Cr-
Dd),\operatorname{Coh}^{1-\theta}(C)[-1])\in\operatorname{Stab}(C).$

######  Remark 5.12.

  1. (1)

If we have $\sigma_{1}=(Z_{1},\operatorname{Coh}^{\theta}(C))$ and
$\sigma_{2}=(Z_{2},\operatorname{Coh}^{\theta}(C))$ with $0\leq\theta<1,$ then
$F_{\sigma_{1}}^{\theta}=F^{\theta}_{\sigma_{2}}.$ Therefore, for any
stability condition $\sigma$ with heart $\operatorname{Coh}^{\theta}(C),$ we
denote $F^{\theta}_{\sigma}$ by $F^{\theta}.$

  2. (2)

Note that
$\mathbb{D}(\operatorname{Coh}^{\theta}(C))\neq\operatorname{Coh}^{1-\theta}(C)[-1].$

  3. (3)

$\sigma^{\vee}_{\mu}=\sigma_{\mu}.$

  4. (4)

$\mu_{\sigma}(E)=-\mu_{{\sigma}^{\vee}}(\mathbb{D}(E)).$

  5. (5)

Let $\sigma=(Z,\operatorname{Coh}^{\theta}(C)),$ then
$\mathbb{D}(F^{\theta})=F^{1-\theta}[-1],$ where $F^{1-\theta}[-1]$ is given
with respect to $\sigma^{\vee}.$

######  Example 5.13.

If $\sigma_{\mu}=(Z_{\mu},\operatorname{Coh}(C)),$ the torsion pair $T^{0}$
and $F^{0}$ is given precisely by the subcategory of torsion sheaves and
torsion-free sheaves respectively. We have that
$\mathbb{D}(\operatorname{Coh}(C))=\langle\mathbb{C}(x)[-1],\mathcal{L}\rangle,$
for any line bundle $\mathcal{L}$ in $\operatorname{Coh}(C)$ and any point
$x\in C.$ Note that $\langle\mathbb{C}(x)[-1],\mathcal{L}\rangle$ is a heart
of a bounded t-structure on $D^{b}(C)$ which does not admit a stability
function. Moreover, we trivially have that $\mathbb{D}(F^{0})=F^{0}.$

We now study $\mathbb{D}_{1}$ on $\mathcal{T}_{C}.$ Let
$\sigma=(Z,\operatorname{TCoh}^{\theta}(C))=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})$
be a pre-stability on $\mathcal{T}_{C}$ with $\sigma_{1}=\sigma_{2}g$ where
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ with $f(0)=0$ and
$T^{-1}=\begin{bmatrix}-A&B\\\ 0&C\end{bmatrix}.$

######  Definition 5.14.

Let
$\sigma=(Z,\operatorname{TCoh}^{\theta}(C))=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2}),$
we define the dual stability condition on $\mathcal{T}_{C}$ as
$\sigma^{*}=\operatorname{gl}_{12}(\sigma_{2}^{\vee},\sigma_{1}^{\vee})=(Z^{*},\operatorname{TCoh}^{1-\theta}(C)[-1]).$

######  Remark 5.15.

  1. (1)

Let $\sigma=(Z,\operatorname{TCoh}^{\theta}(C))$ be a pre-stability condition
as above. If $E=E_{1}\xrightarrow{\varphi}E_{2}$ is $\sigma$-stable with
$\phi_{\sigma}(E)<1,$ i.e. $\Im(Z(E))\neq 0,$ then $E_{1},E_{2}\in
F^{\theta}.$

  2. (2)

Let $E=E_{1}\rightarrow E_{2}\in\operatorname{TCoh}^{\theta}(C),$ with
$E_{1},E_{2}\in F^{\theta}$ then
$\mu_{\sigma}(E)=-\mu_{\sigma*}(\mathbb{D}_{1}(E)).$ Moreover, we also have
that
$\sigma^{*}=\operatorname{gl}_{12}(\sigma_{2}^{\vee},\sigma^{\vee}_{2}gN_{\frac{2B}{C}}).$

  3. (3)

If $\sigma=\operatorname{gl}_{12}(\sigma_{\mu}g,\sigma_{\mu}),$ with
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ where

$T^{-1}=\begin{bmatrix}1&-\alpha\\\ 0&1\end{bmatrix}$

and $f(0)=0,$ then
$\sigma^{*}=\operatorname{gl}_{12}(\sigma_{\mu},\sigma_{\mu}g^{\prime})$ with
$g^{\prime}=gN_{-2\alpha}.$ Note that $g^{\prime}=g^{-1}.$ By the
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action, an object
$E\in\operatorname{TCoh}(C)$ is $\sigma^{*}$-stable if and only if it is
$\sigma^{*}g$-stable, where
$\sigma^{*}g=\operatorname{gl}_{12}(\sigma_{\mu}g,\sigma_{\mu}).$

######  Lemma 5.16.

Let us consider
$\sigma=(Z,\operatorname{TCoh}^{\theta}(C))=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2}).$
An object $E\in\operatorname{TCoh}^{\theta}(C)$ is $\sigma$-stable with
$\phi_{\sigma}(E)<1$ if and only if $\mathbb{D}_{1}(E)$ is $\sigma^{*}$-stable
and $\phi_{\sigma^{*}}(\mathbb{D}_{1}(E))<1.$

######  Proof 36.

As in Remark 5.15 we have that $E_{1},E_{2}\in F^{\theta}.$ By Remark 5.12, it
follows that $\mathbb{D}(E_{1}),\mathbb{D}(E_{2})$ are in $F^{1-\theta}[-1].$
Then
$\mathbb{D}_{1}(E)=\mathbb{D}(E_{2})\rightarrow\mathbb{D}(E_{1})\in\operatorname{TCoh}(C)^{1-\theta}[-1]\textnormal{
and }\phi_{\sigma*}(\mathbb{D}_{1}(E))<1.$ Let us consider $Q=Q_{1}\rightarrow
Q_{2}\in\operatorname{TCoh}^{1-\theta}(C)[-1],$ satisfying $Q_{1},Q_{2}\in
F^{1-\theta}[-1]$ and

$0\rightarrow G\rightarrow\mathbb{D}_{1}(E)\rightarrow Q\rightarrow 0$

a short exact sequence in $\operatorname{TCoh}^{1-\theta}(C)[-1],$ where
$G=G_{1}\rightarrow G_{2}.$ Note that it is enough to prove that
$\phi_{\sigma*}(\mathbb{D}_{1}(E))<\phi_{\sigma*}(Q)$ to show the stability of
$\mathbb{D}_{1}(E).$ Indeed, if $P=P_{1}\rightarrow P_{2}$ is an arbitrary
quotient of $\mathbb{D}_{1}(E),$ either there is an object $Q=Q_{1}\rightarrow
Q_{2}$ with $Q_{1},Q_{2}\in F^{1-\theta}[-1]$ with $P\twoheadrightarrow Q$ and
$\phi_{\sigma*}(Q)<\phi_{\sigma^{*}}(P)$ or $\phi_{\sigma^{*}}(P)=1$ and we
trivially obtain that
$\phi_{\sigma^{*}}(\mathbb{D}_{1}(E))<\phi_{\sigma^{*}}(P).$ We also have that
$G_{1},G_{2}\in F^{1-\theta}[-1]$ as $F^{1-\theta}[-1]$ is closed under
subobjects. Moreover, the duality gives us a correspondence between short
exact sequences on $F^{1-\theta}[-1]$ and $F^{\theta}.$ We obtain a short
exact sequence in $\operatorname{TCoh}^{\theta}(C)$ given by
$0\rightarrow\mathbb{D}_{1}(Q)\rightarrow
E\rightarrow\mathbb{D}_{1}(G)\rightarrow 0.$ By the stability of $E$ we obtain
that $\mu_{\sigma}(\mathbb{D}_{1}(Q))<\mu_{\sigma}(E)$ and it follows if and
only if $\mu_{\sigma*}(\mathbb{D}_{1}(E))<\mu_{\sigma*}(Q).$ As a consequence,
$\mathbb{D}_{1}(E)$ is $\sigma^{*}$-stable.

Let
$\sigma=\operatorname{gl}_{12}(\sigma_{2}g,\sigma_{2})=(Z,\operatorname{TCoh}^{\theta}(C))$
as above and $E=E_{1}\xrightarrow{\varphi}E_{2}$ a $\sigma$-stable object with
$\phi(E)<1,$ $\varphi\neq 0$ and $0<C_{2}r_{2}+Dd_{2}<C_{2}r_{1}+D_{2}d_{1},$
where $[E]=(r_{1},d_{1},r_{2},d_{2}).$ Note that we cannot apply Lemma 5.7
directly.

Note that
$\sigma*=\operatorname{gl}_{12}(\sigma_{2}^{\vee},\sigma_{2}^{\vee}gN_{\frac{2B}{C}}).$
After applying the $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action and
by Lemma 5.16, we have that $E$ is $\sigma$-stable if and only if
$\mathbb{D}_{1}(E)$ is
$\sigma^{\prime}=\sigma^{*}g^{\prime}=(\sigma_{2}^{\vee}g^{\prime},\sigma^{\vee}_{2})$-stable,
where

$g^{\prime}=N_{\frac{-2B}{C}}g^{-1}=(T^{\prime},f^{\prime})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$

and $T^{\prime-1}=\frac{1}{\det(T^{-1})}\begin{bmatrix}C&B\\\
0&-A\end{bmatrix}.$

If
$\sigma_{2}=(Z_{2}(r,d)=A_{2}d_{2}+B_{2}r_{2}+i(C_{2}r_{2}+D_{2}d_{2}),\operatorname{Coh}^{\theta}(C)),$
then

$\sigma^{\vee}_{2}=(Z_{2}^{\vee}(r,d)=A_{2}d_{2}-B_{2}r_{2}+i(C_{2}r_{2}-D_{2}d_{2}),\operatorname{Coh}^{1-\theta}[-1](C)).$

If we apply Lemma 5.7 to $\sigma^{\prime},$ we have that if
$F=F_{1}\xrightarrow{\varphi_{F}}F_{2}$ is $\sigma$-stable with

$C_{2}\operatorname{rk}(F_{2})-D_{2}\deg(F_{2})>C_{2}\operatorname{rk}(F_{1})-D_{2}\deg(F_{1})$

and $[\varphi_{F}]\neq 0,$ then

$\frac{-A_{2}\deg(F_{1})+B_{2}\operatorname{rk}(F_{1})}{C_{2}\operatorname{rk}(F_{1})-D_{2}\deg(F_{1})}\leq\frac{-A_{2}\deg(F_{2})+B_{2}\operatorname{rk}(F_{2})}{C_{2}\operatorname{rk}(F_{2})-D_{2}\deg(F_{2})}.$

Note that $[\mathbb{D}_{1}(E)]=[r_{2},-d_{2},r_{1},-d_{1}].$ As a consequence,
we obtain that $\mathbb{D}_{1}(E)$ satisfies

$C_{2}\operatorname{rk}(\mathbb{D}(E_{1}))-D_{2}\deg(\mathbb{D}(E_{1}))>C_{2}\operatorname{rk}(\mathbb{D}(E_{2}))-D_{2}\deg(\mathbb{D}(E_{2})),$

because $C_{2}r_{2}+Dd_{2}<C_{2}r_{1}+D_{2}r_{1}.$ We can now apply Lemma 5.17
to $\sigma^{\prime}$ and we obtain

$\frac{A_{2}d_{2}+B_{2}r_{2}}{C_{2}r_{2}+D_{2}d_{2}}\leq\frac{A_{2}d_{1}+B_{2}r_{1}}{C_{2}r_{1}+D_{2}d_{1}},$

which is precisely

$\frac{-A_{2}d_{1}-B_{2}r_{1}}{C_{2}r_{1}+D_{2}d_{1}}\leq\frac{-A_{2}d_{2}-B_{2}r_{2}}{C_{2}r_{1}+D_{2}d_{2}}.$

Moreover, we obtain the following result, which gives us the necessary
conditions to have $\sigma$-stable objects.

######  Lemma 5.17.

Let
$\sigma=(Z,\operatorname{TCoh}^{\theta}(C))=\operatorname{gl}_{12}(\sigma,\sigma_{2})$
be a pre-stability condition on $\mathcal{T}_{C}$ with
$Z_{2}(d_{2},r_{2})=A_{2}d_{2}+Br_{2}+i(C_{2}r_{2}+D_{2}r_{2})$ and
$\sigma_{1}=\sigma_{2}g,$ were
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ and

$T^{-1}=\begin{bmatrix}-A&B\\\ 0&C\end{bmatrix}.$

If there is a $\sigma$-semistable object with
$C_{2}r_{1}+D_{2}d_{1}>C_{2}r_{2}+D_{2}d_{2}>0$ then $Cy+Ax\leq-B,$ where
$x=\frac{-A_{2}d_{1}-B_{2}r_{1}}{C_{2}r_{1}+D_{2}d_{1}}$ and
$y=\frac{-A_{2}d_{2}-Br_{2}}{C_{2}r_{2}+D_{2}d_{2}}.$

Moreover, if $[\varphi]\neq 0$ then $y-x\geq 0$ and

$-B\in[Cy+Ax,Ax+Cy+y-x-\frac{C_{2}r_{2}+D_{2}d_{2}}{C_{2}r_{1}+D_{2}d_{1}}y(A+C)\frac{C_{2}r_{1}+D_{2}d_{1}}{C_{2}r_{1}+D_{2}d_{1}-C_{2}r_{2}-D_{2}d_{2}}].$

######  Corollary 5.18.

If $\sigma_{\alpha}$ as defined in Example 3.37 on $\mathcal{T}_{C}$ and $E$
is a $\alpha$-stable object with $[E]=(r_{1}.d_{1},r_{2},d_{2})$ and
$r_{1}\neq r_{2},$ then
$\alpha\in[\frac{d_{2}}{r_{2}}-\frac{d_{1}}{r_{1}},(\frac{d_{2}}{r_{2}}-\frac{d_{1}}{r_{1}})(1+\frac{r_{1}+r_{2}}{|r_{2}-r_{1}|})].$

######  Remark 5.19.

Note that Corollary 5.18 agrees with the necessary conditions for the
existence of $\sigma_{\alpha}$-stable objects [8, Theorem. 6.1].

We will now prove the support property for this type of pre-stability
conditions.

######  Lemma 5.20.

Let $\sigma=(Z,\mathcal{A})=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})$ be
a pre-stability condition, such that there is
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ with
$\sigma_{1}=\sigma_{2}g$ and $T^{-1}=\begin{bmatrix}-A&B\\\ 0&C\\\
\end{bmatrix}.$ Then $\sigma$ satisfies the support property and therefore it
is a Bridgeland stability condition.

######  Proof 37.

First note that we can linearly extend $Z$ to
$\mathcal{N}(\mathcal{T}_{C})\otimes\mathbb{R}\cong\mathbb{R}^{4}.$

We denote $Z_{1}([E])=Z_{1}([i^{*}(E)])$ and $Z_{2}([E])=Z_{2}([j^{!}(E)]).$
As mentioned before, since $i^{*}$ and $j^{!}$ are exact, they induce
homomorphisms on the Grothendieck groups. We use the notation given above for
elements $v\in\mathcal{N}(\mathcal{T}_{C})\otimes\mathbb{R},$ i.e. $Z_{1}(v)$
and $Z_{2}(v).$ We claim that for $\delta=\frac{-CA}{B^{2}}>0,$ the pre-
stability condition $\sigma$ satisfies the support property with the quadratic
form $Q\colon\mathbb{R}^{4}\rightarrow\mathbb{R},$ defined as

$Q(v)=-\Re(Z_{1}(v))\Im(Z_{2}(v)+\Im(Z_{1}(v))\Re(Z_{2}(v))+\Im(Z_{1}(v))\Im(Z_{2}(v))+\delta(\Re(Z_{1}(v))\Re(Z_{2}(v)),$

where $\Re(\alpha)$ and $\Im(\alpha)$ are the real and the imaginary parts
respectively and $\alpha\in\mathbb{R}^{4}.$ Note that under the notation of
equation (5.2), we obtain that

$Q(v)=-Ad_{1}r_{2}-Cd_{2}r_{1}-Br_{1}r_{2}+Cr_{1}r_{2}+\delta(-Ad_{1}d_{2}-Bd_{2}r_{1}).$

We first show that it is negative definite on

$\operatorname{Ker}(Z)=\\{v\in\mathbb{R}^{4}\mid\Re(Z_{1}(v))=-\Re(Z_{2}(v))\textnormal{
and }\Im(Z_{1}(v))=-\Im(Z_{2}(v))\\}.$

In fact, we get that

$\displaystyle Q(v)$ $\displaystyle=$
$\displaystyle\Re(Z_{2}(v))\Im(Z_{2}(v))-\Im(Z_{2}(v))\Re(Z_{2}(v))-\Im(Z_{2}(v))\Im(Z_{2}(v))-\delta(\Re(Z_{2}(v))\Re(Z_{2}(v))$
$\displaystyle=$ $\displaystyle-\Im(Z_{2}(v))^{2}-\delta(\Re(Z_{2}(v))^{2})<0$

if $v\neq 0.$ Note that if $Q(v)=0,$ for $v\in\operatorname{Ker}(Z),$ then
$Z_{i}(v)=0$ for $i=1,2.$ It follows that $v=0,$ as
$\operatorname{rank}(\mathcal{N}(D^{b}(C)))=2.$

Let $E=E_{1}\xrightarrow{\varphi}E_{2}$ be a $\sigma$-stable object. We
consider the following short exact sequence $0\rightarrow j_{*}(E)\rightarrow
E\rightarrow i_{*}(E)\rightarrow 0.$ If $[\varphi]=0,$ Remark 3.14 implies
that either $E_{1}=0$ or $E_{2}=0,$ as $E$ is indecomposable. If $E_{1}=0$ or
$E_{2}=0,$ we clearly have that $Q([E])=0.$

If $\Im(Z_{1}(E))=0,$ it follows that $-\Re(Z_{1}(E))>0.$ We claim that
$\Im(Z_{2}(E))=0.$ Indeed, after considering the decomposition of $E_{2}$ with
respect to the standard torsion pair $(\mathcal{T},\mathcal{F})$ given in
Remark 5.10 on $\mathcal{A},$ we have that there is a subtriple
$T_{2}=0\rightarrow T(E_{2})$ of $E,$ such that $\Im(Z_{2}(T_{2}))=0.$ If
$T_{2}\neq 0,$ then $1=\phi(T_{2})\leq\phi(E)\leq 1,$ which implies that
$\phi(E)=1,$ thus $\Im(Z_{2}(E))=0.$ If $T_{2}=0,$ then $E_{2}\in F$ and the
morphism $\varphi=0,$ as $E_{1}$ is in $T.$ This implies that
$E_{1}\rightarrow 0$ is a subtriple and once again we obtain that

$1=\phi(E_{1}\rightarrow 0)\leq\phi(E)\leq 1.$

So, $\phi(E)=1$ and $\Im(Z_{2}(E))=0$ and this implies that $E=0.$ We obtain
that $\Im(Z_{2}(E))=0$ and $-\Re(Z_{2}(E))>0.$ Hence, we get
$Q(E)=\delta(\Re(Z_{1}(E))\Re(Z_{2}(E)))>0.$

If $\Im(Z_{2}(E))=0,$ then $-\Re(Z_{2}(E))>0.$ By the short exact sequence
above, we obtain that

$1=\phi(0\rightarrow E_{2})\leq\phi(E_{1}\rightarrow 0)\leq 1.$

This implies $\Im(Z_{1}(E))=0$ and $-\Re(Z_{1}(E))>0.$ As a consequence, we
get

$Q(E)=\delta(\Re(Z_{1}(E))\Re(Z_{2}(E)))>0.$

Now we assume $\Im(Z_{1}(E)),\Im(Z_{2}(E))\neq 0$ and therefore $[\varphi]\neq
0$

Let $y=\frac{-\Re(Z_{2}(E)))}{\Im(Z_{2}(E))}$ and
$x=\frac{-\Re(Z_{2}(j_{*}i^{*}(E))}{\Im(Z_{2}(j_{*}i^{*}(E)))}.$ Note that
$\Im(Z_{1}(E))\neq 0,$ implies $\Im(Z_{2}(j_{*}i^{*}(E)))\neq 0.$ After
dividing $Q([E])$ into $\Im(Z_{2}(E))\Im(Z_{2}(j_{*}i^{*}(E)))$ and by
Equation (5.1), we obtain that $Q(v)\geq 0$ if and only if

$-Ax-Cy-B+C+\delta(-Axy-By)\geq 0.$

First note that by the semistability of $E$ and Lemma 5.6 we have that
$Cy+Ax\leq-B$ and $0\leq y-x.$

Our proof falls naturally into two cases:

Case 1: $y\geq 0.$ As $C>0,$ we have $0\leq Cy\leq-Ax-B.$ Then $-Axy-By\geq 0$
and since $-Ax-Cy-B+C\geq 0,$ we get that $Q([E])\geq 0.$

Case 2 If $y<0,$ then $x<y<0.$ Moreover if $-Ax-B\leq 0,$ we have that $-Axy-
By\geq 0,$ and we argue as before. If $0<-Ax-B,$ as $-A>0$ we get
$0<-Ax-B\leq-Ay-B<-B.$ Then $Cy\leq-Ax-B\leq-B$ and $\frac{B}{-A}<y<0.$ Let us
consider the following function

$\displaystyle f\colon(0,-B)\times(\frac{B}{-A},0)$ $\displaystyle\rightarrow$
$\displaystyle\mathbb{R}$ $\displaystyle(x,y)$ $\displaystyle\mapsto$
$\displaystyle\frac{-Ax-Cy-B+C}{-Axy-By}.$

Note $B\leq-Ax-Cy,$ then $0<C\leq-Ax-Cy-B+C.$ Also $\frac{B^{2}}{A}<y(-Ax-B),$
as $xB>0.$ Therefore, we obtain $\frac{-A}{-B^{2}}>\frac{1}{y(x+\alpha)}.$ It
follows that $f(x,y)<\frac{C}{y(x+\alpha)}<\frac{-CA}{-B^{2}}.$ Since
$\delta=\frac{-CA}{B^{2}},$ we have that $f(x,y)<-\delta.$ Hence, we get $-Ax-
Cy-B+C+\delta(-Axy-By)>0.$

#### Support property for CP-glued pre-stability conditions with negative
discriminant

Let $\sigma=(Z,\mathcal{A})=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})$ be
a pre-stability condition, where $\sigma_{1}=(Z_{1},\mathcal{A}_{1})$ and
$\sigma_{2}=(Z_{2},\mathcal{A}_{2}),$ such that there is
$g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ with
$\sigma_{1}=\sigma_{2}g$ where $(T,f)$ satisfies that $M\coloneqq
T^{-1}=\begin{bmatrix}-A&B\\\ -D&C\\\ \end{bmatrix}$ with

$\Delta(M)=\operatorname{Tr}(M)^{2}-4\det(M)=(A+C)^{2}-4BD<0\textnormal{ and
}0<f(0)<1.$

######  Remark 5.21.

If $E=E_{1}\xrightarrow{\varphi}E_{2}$ is a $\sigma$-semistable object with
$\Im(Z_{2}(E_{1}))<0.$ Then $E_{1}\in\mathcal{A}_{2}[1].$

######  Lemma 5.22.

Let $\sigma=(Z,\mathcal{A})$ be as above. Then $\sigma$ is a Bridgeland
stability condition.

######  Proof 38.

We just need to prove that it satisfies the support property. Note that we can
extend $Z$ to
$\mathcal{N}(\mathcal{T}_{\mathcal{A}})\otimes\mathbb{R}\cong\mathbb{R}^{4}.$
We claim that $\sigma$ satisfies the support property with respect to the
quadratic form $Q\colon\mathbb{R}^{4}\rightarrow\mathbb{R}$ defined as

$Q(v)=-\Im(Z_{2}(j^{!}(v))))\Re(Z_{2}(i^{*}(v)))+\Im(Z_{2}(i^{*}v)))\Re(Z_{2}(j^{!}(v)).$

We use the following notation

$d_{1}=-\Re(Z_{2}(i^{*}(v)))\textnormal{ and }r_{1}=\Im(Z_{2}(i^{*}(v))),$
$d_{2}=-\Re(Z_{2}(j^{!}(v)))\textnormal{ and }r_{2}=\Im(Z_{2}(j^{!}(v))).$

Note that $r_{2}\geq 0.$ We first show that $Q$ is negative definite on

$\operatorname{Ker}(Z)=\\{v\in\mathbb{R}^{4}\mid\Re(Z_{2}(j^{!}v))=-\Re(Z_{1}(i^{*}v))\textnormal{
and }\Im(Z_{2}(i^{*}v))=-\Im(Z_{1}(j^{!}v))\\}.$

Note that
$Z_{1}(w)=-A\Re(Z_{2}(w))+B\Im(Z_{2}(w))+i(-D\Re(Z_{2}(w))+C\Im(Z_{2}(w))),$
with $w\in\mathbb{R}^{2}.$ We get that
$Q(v)=-Dd_{1}^{2}+(A+C)d_{1}r_{1}+Br_{1}^{2}<0.$ Indeed, as $0\leq f(0)=t<1$
and $t<1<t+1,$ we have $D>0.$ Moreover since $\Delta(M)=(A+C)^{2}-4BD<0,$ then
$B>0$ and $-Dd^{2}-(A+C)dr-Br^{2}<0$ for all $(r,d)\in\mathbb{R}^{2}.$ If
$(r,d)=0,$ we obtain that $v=0.$

Now let $E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{A}$ be a $\sigma$-stable
object. We show that $Q([E])\geq 0.$ First of all, if
$[E]=v\in\mathcal{N}(\mathcal{T}_{\mathcal{A}})$ with $\Im(Z_{2}(v))=0,$ then
$\Re(Z_{2}(v))<0$ and after considering the following short exact sequence
$j_{*}(E_{2})\rightarrow E\rightarrow i_{*}(E_{1})\rightarrow j_{*}(E_{2})[1]$
in $\mathcal{A},$ we obtain

$1=\phi(0\rightarrow E_{2})\leq\phi(E_{1}\rightarrow 0)\leq 1.$

Then $\phi(E_{1}\rightarrow 0)=1,$ and this implies that $Cr_{1}+Dd_{1}=0$ and
$r_{1}\leq 0.$ As $-d_{2}<0,$ we get

$Q([E])=-d_{2}r_{1}\geq 0.$

If $r_{1}=0,$ as $Cr_{1}+Dd_{1}\geq 0,$ then $Dd_{1}\geq 0$ and $d_{1}\geq 0.$
Therefore, we have $Q([E])=d_{1}r_{2}\geq 0.$

Let us now assume that $r_{2},r_{1}\neq 0.$ From the short exact sequence
mentioned above and the correspondence between slope and phase, we obtain

$\frac{d_{2}}{r_{2}}\leq\frac{-Ad_{1}+Br_{1}+d_{2}}{Cr_{1}+Dd_{1}+r_{2}}\textnormal{
i.e.\ }-Dd_{1}d_{2}-Cr_{1}d_{2}-Ad_{1}r_{2}-Br_{1}r_{2}\geq 0.$

We define $x=\frac{d_{1}}{r_{1}}$ and $y=\frac{d_{2}}{r_{2}}.$ Let us consider
two cases:

Case 1: $r_{1}>0.$ Since by definition $Cr_{1}+Dd_{1}\geq 0,$ then $C+Dx\geq
0.$ Moreover we have $C+Dx>0.$ Indeed, if $C+Dx=0,$ we get
$E_{1}\in\mathcal{A}_{2}[1]$ and $r_{1}\leq 0,$ which contradicts our
assumption. Then, we obtain $y\leq\frac{-Ax-B}{Dx+C}.$

As $\Delta(M)<0$ and $D>0,$ then for all $t\in\mathbb{R}$ we have that
$Dt^{2}+(A+C)t+B>0.$ Thus, we obtain $x>\frac{-Ax+B}{Dx+C}.$ Finally, we get
$x>y$ or equivalently $Q([E])=d_{1}r_{2}-d_{2}r_{1}>0.$

Case 2: $r_{1}<0.$ We claim that $[\varphi]=0.$ Indeed, by Remark 5.21, we
have that $E_{1}\in\mathcal{A}_{2}[1].$ Then
$\varphi\in\operatorname{Hom}_{D^{b}(\mathcal{A})}(E_{1},E_{2})=0,$ as
$\mathcal{A}_{2}$ is the heart of a bounded t-structure in
$D^{b}(\mathcal{A}).$ This implies that $E_{1}=0$ or $E_{2}=0,$ and as a
consequence, $Q([E])=0.$

We will now prove that all CP-glued pre-stability conditions on
$\mathcal{T}_{C}$ satisfy the support property.

######  Proposition 5.23.

If the pair $\sigma=(Z,\mathcal{A})$ on $\mathcal{T}_{C}$ is a pre-stability
condition with ${\sigma=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2}),}$ then
it satisfies the support property and therefore, it is a Bridgeland stability
condition.

######  Proof 39.

There is ${g=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})}$ with
${(Z_{1},\mathcal{A}_{1})=\sigma_{1}=\sigma_{2}g}$ and
$\sigma_{2}=(Z_{2},\mathcal{A}_{2}).$ The proof falls naturally into the
following cases:

Case 1: If $f(0)\geq 1,$ then this follows directly from Lemma 5.1.

Case 2: If $0\leq f(0)<1$ and $\Delta(M)\geq 0,$ then we have the existence of
real eigenvalues $\lambda_{1},\lambda_{2}\in\mathbb{R}.$ If
$\lambda_{1},\lambda_{2}>0,$ then by Lemma 5.20 and the fact that the support
property is stable under the
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action, we have that $\sigma$
has the support property. If $\lambda_{1},\lambda_{2}<0,$ then by arguments
along the lines of Proposition 4.43 there is
$h\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ such that $\sigma
h=\operatorname{gl}_{12}(\sigma_{1}h,\sigma_{2}h)$ with
$\sigma_{i}h=(Z^{\prime}_{i},\mathcal{A}^{\prime}_{i})$ and
$\operatorname{Hom}^{\leq
1}(i_{*}\mathcal{A}^{\prime}_{1},j_{*}\mathcal{A}^{\prime}_{2})=0,$ therefore
it follows from Lemma 5.1.

Case 3: If $0<f(0)<1$ and $\Delta(M)<0,$ then this case follows directly from
Lemma 5.22.

######  Remark 5.24.

By Remark 2.46, after applying Serre duality, we have that any pre-stability
condition $\sigma\in\Theta_{i},$ with $i\in\\{1,2,3\\}$ satisfies the support
property. By Theorem 4.47, we just need to prove the support property for
$\sigma\in\Gamma.$

We will now prove the support property for $\sigma\in\Gamma$ just when
$g(C)=1.$ For $g(C)>1,$ we conjecture that it is also satisfied. We start by
studying the $\sigma$-semistable objects in the pre-stability conditions in
Lemma 3.48 under the assumption that $g(C)\geq 1.$

#### Semistability on non-gluing pre-stability conditions

We will now study $\sigma$-semistable objects, where $\sigma=(Z,\mathcal{A})$
satisfies that $i_{*}(\mathbb{C}(x)),l_{*}(\mathbb{C}(x))$ are $\sigma$-stable
and $j_{*}(\mathbb{C}(x))$ is $\sigma$-stable of phase one. After applying the
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action on $\sigma$ the objects
$i_{*}(\mathbb{C}(x)),$ $j_{*}(\mathbb{C}(x))$ and $l_{*}(\mathbb{C}(x))$ are
always $\sigma$-stable. We use the description of the hearts given in Lemma
4.18, Lemma 4.19 and Lemma 4.20 to prove that all $\sigma$-semistable objects
have a particular form.

######  Lemma 5.25.

If $E=E_{1}\xrightarrow{\varphi}E_{2}\in\mathcal{A}$ is $\sigma$-semistable
then $E\in\operatorname{TCoh}(C)$ or $E\in\mathcal{H}_{23}[-1]$ or
$E\in\mathcal{H}_{31}[-1].$

######  Proof 40.

First note that there are elements
$g_{1},g_{2}\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ with
$\delta_{i}=\sigma g_{i}=(W_{i},\mathcal{B}_{i})$ for $i=1,2,$ such that
$\delta_{1}$ satisfies that $i_{*}(\mathbb{C}(x))$ is $\delta_{1}$-stable of
phase one, $j_{*}(\mathbb{C}(x)),l_{*}(\mathbb{C}(x))$ are $\delta_{1}$-stable
and $\delta_{2}$ satisfies that $l_{*}(\mathbb{C}(x))$ is $\delta_{2}$-stable
of phase one and $j_{*}(\mathbb{C}(x)),i_{*}(\mathbb{C}(x))$ are
$\delta_{2}$-stable. We have $E\in\mathcal{A}.$ Note that Lemma 4.18 describes
the cohomology of $E.$ For an object
$G=G_{1}\xrightarrow{\psi}G_{2}\in\mathcal{B}_{1},$ Lemma 4.19 describes its
cohomology and if $G=G_{1}\xrightarrow{\psi}G_{2}\in\mathcal{B}_{2},$ in Lemma
4.20 we obtain the description of its cohomology.

As $E$ is $\delta_{i}$-semistable we have that $E\in\mathcal{B}_{i}[n],$ for
$i=1,2$ and $n\in\mathbb{Z},$ where the only possible cases are $n=1,0,-1,-2.$
We study the non-trivial cases.

  1. (1)

If $E\in\mathcal{A}\cap\mathcal{B}_{1}[-2]\cap\mathcal{B}_{2}[-1],$ as the
intersection is contained in $\operatorname{Coh}_{3}(C)[-1],$ then
$E\in\mathcal{H}_{23}[-1].$

  2. (2)

If $E\in\mathcal{A}\cap\mathcal{B}_{1}[-1]\cap\mathcal{B}_{2}[-1],$ then
$E\in\mathcal{H}_{23}[-1].$

  3. (3)

If $E\in\mathcal{A}\cap\mathcal{B}_{1}[-1]\cap\mathcal{B}_{2},$ then
$E\in\mathcal{H}_{31}[-1].$

  4. (4)

If $E\in\mathcal{A}\cap\mathcal{B}_{1}\cap\mathcal{B}_{2}[-1],$ as the
intersection is contained in $\operatorname{Coh}_{2}(C),$ then
$E\in\operatorname{TCoh}(C).$

  5. (5)

If $E\in\mathcal{A}\cap\mathcal{B}_{1}\cap\mathcal{B}_{2},$ then
$E\in\operatorname{TCoh}(C).$

  6. (6)

If $E\in\mathcal{A}\cap\mathcal{B}_{1}\cap\mathcal{B}_{2}[1],$ as the
intersection is contained in $\operatorname{Coh}_{1}(C),$ then
$E\in\operatorname{TCoh}(C).$

###  5.3. Support property for non-gluing pre-stability conditions with
negative discriminant and $g=1$

We consider pre-stability conditions $\sigma=(Z,\mathcal{A})$ as constructed
in Lemma 3.48 with $\Delta<0,$ i.e. there is
$\sigma_{1}=(Z_{1},\operatorname{Coh}_{1}^{r_{1}}(C))=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$
such that $\operatorname{Coh}_{1}^{r_{1}}(C)\subseteq\mathcal{A}$ and
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\operatorname{Coh}_{1}^{r_{1}}(C)}}=Z_{1},$
with $-1<f(0)<0.$ We assume $\Delta(M)<0,$ where $M=T^{-1}.$ Under the
assumption that $g(C)=1,$ these pre-stability conditions satisfy the support
property and as a consequence, they are Bridgeland stability conditions.

Since $\Delta(M)<0,$ after applying the
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-action we never obtain a CP-
glued pre-stability condition, because $f(\theta)<\theta$ for all
$\theta\in\mathbb{R}.$ However, under the assumption $g=1,$ the quadratic form
induced by the Euler bilinear form $-\chi(E,E)=d_{2}r_{1}-d_{1}r_{2}$ is
negative definite on $\operatorname{Ker}(Z)$ as in Lemma 5.22. Therefore, it
is a good candidate for being a quadratic form appearing in the support
property.

We will now prove some useful statements about $\mathcal{A}.$

######  Lemma 5.26.

If $F\in\operatorname{Coh}(C)$ is $\mu$-stable, then $i_{*}(F),$ $j_{*}(F)$
and $l_{*}(F)$ are $\sigma$-stable.

######  Proof 41.

The proof goes along the lines of Proposition 4.5. If $F$ is $\mu$-stable, it
is either a skyscraper sheaf of a torsion-free sheaf. In Proposition 4.5 we
proved the statement for skyscraper sheaves, therefore we assume that $F$ is
torsion-free and $r>0.$ Let us assume that $i_{*}(F)$ is not
$\sigma$-semistable. Then, we consider the last triangle of its
Harder–Narasimhan filtration $E\rightarrow i_{*}(F)\rightarrow A\rightarrow
E[1]$ which satisfies $\operatorname{Hom}_{\mathcal{T}_{C}}^{\leq n}(E,A)=0,$
with $n\leq 0.$ By Lemma 4.1 we have that
$\operatorname{Hom}_{D^{b}(C)}(E_{1},A_{1})=0$ and
$E_{1},A_{1}\in\operatorname{Coh}(C).$ By Serre duality on $C$, we get
$\operatorname{Hom}_{D^{b}(C)}(A_{1},E[1])=0,$ which implies that the short
exact sequence

$0\rightarrow E_{1}\rightarrow F\rightarrow A_{1}\rightarrow 0$

splits. As $F$ is $\mu$-stable, it is indecomposable. Therefore, either
$E_{1}=0$ or $A_{1}=0.$ Following exactly the same steps as in Lemma 4.5 with
$X=F,$ we show that $A_{1}=0$ and $j_{*}(F),l_{*}(F)$ are $\sigma$-stable. As
in Lemma 4.7, the last triangle of the HN-filtration of $i_{*}(F)$ is given by

$l_{*}(F)\rightarrow i_{*}(F)\rightarrow j_{*}(F)[1]\rightarrow l_{*}(F)[1]$

and this implies that $\phi(l_{*}(F))>\phi(j_{*}(F))+1.$ As
$j_{*}(F)\in\operatorname{Coh}_{2}(C)\subseteq\mathcal{A},$ then
$1<\phi(j_{*}(F))+1\leq 2,$ which implies that $1<\phi(l_{*}(F)).$ Moreover,
note that $\phi(l_{*}(F))<2.$ By the stability of $l_{*}(\mathbb{C}(x))$ we
have that $1<\phi_{4}<2$ and we also have a non-zero morphism
$l_{*}(F)\rightarrow l_{*}(\mathbb{C}(x)).$

By Lemma 3.50 we have that
$\operatorname{Coh}^{r_{3}}(C)\subseteq\mathcal{A},$ so that
$l_{*}(F)[-1]\in\mathcal{A}.$ Moreover, by the correspondence between slope
and phase, we have that

$\phi(j_{*}(F))=\frac{d}{r}<\frac{Ad+Br-d}{-Cr-Dd-r}=\phi(l_{*}(F)[-1]),$

which implies $-Dd^{2}-(A+C)dr-Br^{2}<0$ and induces a contradiction, because
$\Delta(M)=(A+C)^{2}-4BD<0$ and $D,B<0.$ Therefore, we obtain that $i_{*}(F)$
is $\sigma$-semistable. Now we assume that $i_{*}(F)$ is strictly-semistable,
by Remark 4.10 we obtain exactly the same contradiction. Therefore, we get
that $i_{*}(F)$ is $\sigma$-stable. Analogously, we prove that $j_{*}(F)$ and
$l_{*}(F)$ are $\sigma$-stable.

######  Lemma 5.27.

If $F\in\operatorname{Coh}(C)$ is $\mu$-semistable, then $i_{*}(F),$
$j_{*}(F)$ and $l_{*}(F)$ are $\sigma$-semistable.

######  Proof 42.

If $F=\mathbb{C}(x),$ the statement is already proved by Lemma 4.16.
Therefore, we assume that $F$ is torsion-free. Let us consider a JH-filtration
of $F$ with respect to $\mu.$ Note that all the $\mu$-stable factors $A_{i},$
for $i=0,\cdots,n,$ have the same slope $\mu(F).$ By Lemma 5.26, we obtain
that $j_{*}(A_{i})$ is $\sigma$-stable in $\mathcal{A}.$ As
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\operatorname{Coh}_{2}(C)}}=Z_{\mu},$
we have that $\phi(j_{*}(A_{i}))=\phi(j_{*}(F))=\lambda,$ with
$\lambda\in\mathbb{R}.$ Since the category $\mathcal{P}(\lambda)$ is closed
under extensions, we obtain that $j_{*}(F)$ is $\sigma$-semistable. Note that
since $F$ is $\mu$-semistable, then $i_{*}(F)$ is in $\mathcal{A}$ or in
$\mathcal{A}[1]$ and $l_{*}(F)$ is in $\mathcal{A}$ or in $\mathcal{A}[1].$
Analogously, the same conclusion can be drawn for $i_{*}(F)$ and $l_{*}(F).$

######  Remark 5.28.

Note that $\mathcal{A}\cap\mathcal{C}_{2}=\operatorname{Coh}_{2}(C).$

######  Lemma 5.29.

Let $E=E_{1}\xrightarrow{\varphi}E_{2}\in\operatorname{TCoh}(C)$ be a
$\sigma$-semistable in $\mathcal{A}$. Then
$\operatorname{Hom}_{\mathcal{T}_{C}}(E,E[2])=0.$

######  Proof 43.

First note that $E\in\operatorname{TCoh}(C)\cap\mathcal{A}=\mathcal{F},$ where
$\mathcal{A}=(\mathcal{F},\mathcal{T}[-1])$ as in Lemma 4.18. By Serre duality
$\operatorname{Hom}_{\mathcal{T}_{C}}(E,E[2])=\operatorname{Hom}_{\mathcal{T}_{C}}(E[2],\mathcal{S}_{\mathcal{T}_{C}}(E))^{*}=\operatorname{Hom}(E[1],E_{2}\rightarrow
C(\varphi))^{*}.$ It suffices to prove that
$\operatorname{Hom}(E[1],E_{2}\rightarrow C(\varphi))=0.$

Let us consider the triangle
$i_{*}(C(\varphi))\rightarrow\mathcal{S}_{\mathcal{T}_{C}}(E)[-1]\rightarrow
j_{*}(E_{2})\rightarrow i_{*}(C(\varphi))[1].$ Applying $\operatorname{Hom}$
induces a long exact sequence and as a consequence, it is enough to prove that
$\operatorname{Hom}_{\mathcal{T}_{C}}(E[1],i_{*}(E_{2}))=0\textnormal{ and
}\operatorname{Hom}_{\mathcal{T}_{C}}(E[1],j_{*}(C(\varphi)))=0.$

We have
$\operatorname{Hom}_{\mathcal{T}_{C}}(E[1],i_{*}(E_{2}))=\operatorname{Hom}_{D^{b}(C)}(E_{1}[1],E_{2})=0$
as $E_{1}\textnormal{, }E_{2}\in\operatorname{Coh}(C).$ We will now prove that
$\operatorname{Hom}_{\mathcal{T}_{C}}(E[1],j_{*}(C(\varphi)))=0.$

Case 1: If $\operatorname{Ker}(\varphi)=0,$ we obtain that
$C(\varphi)=\operatorname{Coker}(\varphi)$ and by adjointness we get

$\operatorname{Hom}_{\mathcal{T}_{C}}(E[1],j_{*}(C(\varphi)))=\operatorname{Hom}_{D^{b}(C)}(\operatorname{Coker}(\varphi)[1],\operatorname{Coker}(\varphi))=0.$

Case 2: $\operatorname{Ker}(\varphi)\neq 0.$ It is enough to show that
$\phi(E)+1>\phi^{+}(j_{*}(C(\varphi))).$ Let us compute
$\phi^{+}(j_{*}(C(\varphi))).$ By Remark 5.28 we have that
$\mathcal{A}\cap\mathcal{C}_{2}=\operatorname{Coh}_{2}(C),$ so
$j_{*}(C(\varphi))\notin\mathcal{A}.$ As a consequence, we first need to
consider its filtration in the t-structure induced by $\mathcal{A},$ given by

${0}$${j_{*}(\operatorname{Ker}(\varphi))[1]}$${j_{*}(C(\varphi))}$${j_{*}(\operatorname{Ker}(\varphi))[1]}$${j_{*}(\operatorname{Coker}(\varphi)).}$

By definition,
$\phi^{+}(j_{*}(C(\varphi)))=\phi^{+}(j_{*}(\operatorname{Ker}(\varphi)))+1.$
Let us consider the HN-filtration $0\subseteq H_{1}\cdots\subseteq
H_{n-1}\subseteq H_{n}=j_{*}(\operatorname{Ker}(\varphi))$ of
$j_{*}(\operatorname{Ker}(\varphi))$ in $\mathcal{A}$ with respect to
$\sigma.$ Note that by Lemma 5.27, if $F\in\operatorname{Coh}(C)$ is
$\mu$-semistable then $j_{*}(F)$ is also $\sigma$-semistable. Therefore if we
consider the HN-filtration of $\operatorname{Ker}(\varphi)$ with respect to
$\mu,$ as $\operatorname{Coh}_{2}(C)\subseteq\mathcal{A}$ and
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\operatorname{Coh}_{2}(C)}}=Z_{\mu},$
it will give us the HN-filtration of $j_{*}(\operatorname{Ker}(\varphi))$ in
$\mathcal{A}$ with respect to $\sigma.$ By the uniqueness of the HN-
filtration, we deduce that $H_{i}\in\operatorname{Coh}_{2}(C),$ for all
$i=0,\dots,n.$ Moreover, we have that $H_{1}\neq 0$ is $\sigma$-semistable and
$\phi(H_{1})=\phi^{+}(j_{*}(\operatorname{Ker}(\varphi)))=\phi^{+}(j_{*}(C(\varphi)))-1.$

Let $H_{1}=0\rightarrow F_{1},$ with $F_{1}\in\operatorname{Coh}(C).$ By
definition $F_{1}\subseteq\operatorname{Ker}(\varphi).$ As
$\operatorname{Ker}(\varphi)\rightarrow 0$ is a subobject of $E$ in
$\operatorname{TCoh}(C)$ and $\mathcal{F}$ is closed under subobjects, we
obtain that $F_{1}\rightarrow 0\in\mathcal{F}\subseteq\mathcal{A}.$ By Lemma
5.27, as $F_{1}$ is $\mu$-semistable, $i_{*}(F_{1})$ is also
$\sigma$-semistable. Moreover, we have a non-zero morphism
$i_{*}(F_{1})\rightarrow E.$ As they are both $\sigma$-semistable this implies
that $\phi_{\sigma}(i_{*}(F_{1}))\leq\phi_{\sigma}(E).$

Let $d=\deg(F_{1})$ and $r=\operatorname{rank}(F_{1}).$ By the definition of
$\mathcal{F},$ we get that $\operatorname{Ker}(\varphi)$ and therefore $F_{1}$
is torsion-free and $r>0.$ As $i_{*}(F_{1})\in\mathcal{F},$ we also have that
$Cr+Dd\geq 0.$ Moreover, we have $\phi(H_{1})<\phi(i_{*}(F_{1})).$ As a
consequence, we obtain
$\phi^{+}(j_{*}(C(\varphi)))-1=\phi(j_{*}(F_{1}))<\phi(E)$ as we wanted to
prove.

We now use Serre duality to obtain the same results for the $\sigma$-stable
objects $E\in\mathcal{H}_{23}[-1],$ $\mathcal{H}_{31}[-1].$

######  Corollary 5.30.

If $E$ is $\sigma$-stable, then
$\operatorname{Hom}_{\mathcal{T}_{C}}(E,E[2])=0.$

Now we compute the Euler form for all $\sigma$-stable objects.

######  Lemma 5.31.

If $E$ is a $\sigma$-stable object, then $-\chi(E,E)=d_{2}r_{1}-d_{1}r_{2}\geq
0.$

######  Proof 44.

The proof falls naturally into two cases:

Case 1: $[\varphi]=0.$ It implies that either $E_{1}=0$ or $E_{2}=0.$ If not
it would contradict that $E$ is $\sigma$-stable. It clearly follows that
$-\chi(E,E)=0.$

Case 2: $[\varphi]\neq 0.$ As $E\in\mathcal{A}$ and $\mathcal{A}$ is the heart
of a bounded t-structure, we have that
$\operatorname{Hom}_{\mathcal{T}_{C}}(E,E[n])=0$ for all $n<0.$ By Corollary
5.30 we have that $\operatorname{Hom}_{\mathcal{T}_{C}}(E,E[2])=0$ and as
$\operatorname{TCoh}(C)$ has homological dimension $2,$ we get that
$\mathcal{H}_{23}[-1]$ and $\mathcal{H}_{31}[-1]$ also have homological
dimension 2. Therefore, it follows that
$\operatorname{Hom}_{\mathcal{T}_{C}}(E,E[n])=0$ for $n\geq 2.$ As a
consequence, we obtain
$-\chi(E,E)=-\hom_{\mathcal{T}_{C}}(E,E)+\hom_{\mathcal{T}_{C}}(E,E[1]).$

As $E$ is $\sigma$-stable, it follows that $-\hom_{\mathcal{T}_{C}}(E,E)=-1.$
To prove our claim, it suffices to show that
$\hom_{\mathcal{T}_{C}}(E,E[1])>0.$ By Serre duality
$\operatorname{Hom}_{\mathcal{T}_{C}}(E,E[1])=\operatorname{Hom}(E[1],\mathcal{S}_{\mathcal{T}_{C}}(E))^{*}$
where
$\mathcal{S}_{\mathcal{T}_{C}}(E)=E_{2}[1]\xrightarrow{i_{E}[1]}C(\varphi)[1].$
We have $\hom_{\mathcal{T}_{C}}(E,\mathcal{S}_{\mathcal{T}_{C}}(E))>0$ because
there is a non-zero morphism
$E\rightarrow\mathcal{S}_{\mathcal{T}_{C}}(E)[-1].$ Therefore,
$-\chi(E,E)=d_{2}r_{1}-d_{1}r_{2}>0.$

######  Proposition 5.32.

Let $\sigma=(Z,\mathcal{A})$ be a pre-stability condition as in Lemma 3.48
with $\Delta(M)<0.$ Then it satisfies the support property and therefore it is
a Bridgeland stability condition.

######  Proof 45.

We claim that $\sigma$ satisfies the support property with respect to the
following quadratic form $Q(r_{1},d_{1},r_{2},d_{2})=d_{2}r_{1}-d_{1}r_{2}.$
We first show that $Q$ is negative definite on
$\operatorname{Ker}(Z)=\\{(r_{1},d_{1},r_{2},d_{2})\mid
d_{2}=Ad_{1}+Br_{1}\textnormal{ and }r_{2}=-Cr_{1}-Dd_{1}\\}.$ Let
$(r_{1},d_{1},r_{2},d_{2})\in\operatorname{Ker}(Z),$ then
$Q(r_{1},d_{1},r_{2},d_{2})=Dd_{1}^{2}+(A+C)d_{1}r+Br_{1}^{2}<0$ as
$-1<f(0)=r<0,$ we have that $D<0.$ Moreover since $\Delta(M)=(A+C)^{2}-4BD<0,$
then $B<0$ and $Dd_{1}^{2}+(A+C)d_{1}r_{1}+Br_{1}^{2}<0$ for all
$(r_{1},d_{1})\in\mathbb{R}^{2}.$

Let $E=E_{1}\xrightarrow{\varphi}E_{2}$ be a $\sigma$-semistable object. By
[5, Lemma A.6] it is enough to show that $Q(E)\geq 0$ for $\sigma$-stable
objects. By Claim 5.31 we have that $d_{2}r_{1}-d_{1}r_{2}\geq 0.$

######  Remark 5.33.

To prove the support property of the pre-stability conditions with negative
discriminant in the case $g>1,$ we would need to prove Lemma 5.29. But this
would not be enough as we cannot use the Euler form
$-\chi(E,E)=d_{2}r_{1}-d_{1}r_{2}-(1-g)(r_{1}^{2}+r_{2}^{2}-r_{1}r_{2})$
directly, because it is not negative definite on $\operatorname{Ker}(Z).$

######  Theorem 5.34.

Let $g=1$ and $\sigma\in\Theta_{12}$ be a pre-stability condition. Then it
satisfies the support property and therefore is a Bridgeland stability
condition.

######  Proof 46.

If $\sigma\in\Theta_{i},$ this follows directly from Proposition 5.23 and if
$\sigma\in\Gamma,$ then it follows from Proposition 5.32.

##  6\. Topological description of $S_{12}$

It is now our purpose to study the topology of
$\operatorname{Stab}(\mathcal{T}_{C}),$ we proceed by defining the following
sets

$\displaystyle S_{12}$ $\displaystyle=$
$\displaystyle\\{\sigma\in\operatorname{Stab}(\mathcal{T}_{C})$
$\displaystyle\mid$ $\displaystyle
i_{*}(\mathbb{C}(x)),j_{*}(\mathbb{C}(x)),i_{*}(\mathcal{O}_{C}),j_{*}(\mathcal{O}_{C})\>\>\sigma\textnormal{-stable}\textnormal{
for all closed points $x\in C$}$ $\displaystyle\\}$ , $\displaystyle S_{23}$
$\displaystyle=$
$\displaystyle\\{\sigma\in\operatorname{Stab}(\mathcal{T}_{C})$
$\displaystyle\mid$ $\displaystyle
j_{*}(\mathbb{C}(x)),l_{*}(\mathbb{C}(x)),j_{*}(\mathcal{O}_{C}),l_{*}(\mathcal{O}_{C})\>\>\sigma\textnormal{-stable}\textnormal{
for all closed points $x\in C$}$ $\displaystyle\\}$ , $\displaystyle S_{31}$
$\displaystyle=$
$\displaystyle\\{\sigma\in\operatorname{Stab}(\mathcal{T}_{C})$
$\displaystyle\mid$ $\displaystyle
i_{*}(\mathbb{C}(x)),l_{*}(\mathbb{C}(x)),i_{*}(\mathcal{O}_{C}),l_{*}(\mathcal{O}_{C})\>\>\sigma\textnormal{-stable}\textnormal{
for all closed points $x\in C$}$ $\displaystyle\\}$ ,

Throughout the whole section, we assume that every pre-stability condition
$\sigma\in\Gamma$ is also a Bridgeland stability condition, i.e. it satisfies
the support property. The aim of this section is to prove that $S_{12}$ is an
open, connected four dimensional complex manifold. It is based on the well-
behaved wall and chamber decompositions of the space of stability conditions.
See [3, Proposition 3.3] or [12, Proposition 9.3]. Let us consider a connected
component $\operatorname{Stab}^{\dagger}(\mathcal{T}_{C})$ of
$\operatorname{Stab}(\mathcal{T}_{C}).$

######  Lemma 6.1.

The set $S_{12}\cap\operatorname{Stab}^{\dagger}(\mathcal{T}_{C})$ is open in
$\operatorname{Stab}^{\dagger}(\mathcal{T}_{C}).$

######  Proof 47.

Let
$S=\\{i_{*}(\mathbb{C}(x)),j_{*}(\mathbb{C}(x)),i_{*}(\mathcal{O}_{C}),j_{*}(\mathcal{O}_{C})\mid
x\in C\\}\subseteq\mathcal{T}_{C}.$ First note that the classes of
$i_{*}(\mathbb{C}(x)),$ $j_{*}(\mathbb{C}(x)),$ and $i_{*}(\mathcal{O}_{C})$
and $j_{*}(\mathcal{O}_{C})$ in $K(\mathcal{T}_{C})$ are primitive. By [3,
Proposition 3.3], we have a well-behaved wall and chamber decomposition. We
consider the set $\Theta^{\dagger}$ of points
$\sigma\in\operatorname{Stab}^{\dagger}(\mathcal{T}_{C})$ at which all objects
of $S$ are $\sigma$-stable. We will now prove that $\Theta^{\dagger}$ is open.
Let $B\subseteq\operatorname{Stab}^{\dagger}(\mathcal{T}_{C})$ be a compact
set, we show that $F=\\{\sigma\in B\mid\textnormal{ not every }E\in
S\textnormal{ is stable in }\sigma\\}$ is a closed set. As in [12, Proposition
9.4] this follows from the fact that $F=\cup^{n}_{j=0}\bar{C_{j}},$ where each
$C_{j}$ is a chamber in which some $E\in S$ is not stable.

Let $g_{1},g_{2}\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}),$ with
$g_{1}=(T_{1},f_{1})$ and $g_{2}=(T_{2},f_{2}).$ We denote by

$M_{i}=T_{i}^{-1}=\begin{bmatrix}-A_{i}&B_{i}\\\
-D_{i}&C_{i}\end{bmatrix}\textnormal{ for }i=1,2.$

Let us consider the subset

$\mathcal{P}_{12}=\\{(\sigma_{1},\sigma_{2})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})^{2}\mid\phi_{0}<\phi_{2}+1,\phi_{1}<\phi_{3}+1\textnormal{
and if }\phi_{0}>\phi_{2}\textnormal{, then }\det(M_{1}+M_{2})>0\\},$

where $f_{1}(0)=n+\theta_{1},$ $f_{2}(0)=m+\theta_{2}$, $n,m\in\mathbb{Z}$ and
$\theta_{1},\theta_{2}\in[0,1),$
$\rho(\sigma_{1})=(m_{0},m_{1},\phi_{0},\phi_{1})$ and
$\rho(\sigma_{2})=(m_{2},m_{3},\phi_{2},\phi_{3}),$ as we explained in Remark
2.53.

######  Remark 6.2.

For $\sigma\in S_{12},$ we consider

$\displaystyle\phi_{0}=\phi(i_{*}(\mathbb{C}(x)))$ and
$\displaystyle\phi_{1}=\phi(i_{*}(\mathcal{O}_{C}))$ (6.1)
$\displaystyle\phi_{2}=\phi(j_{*}(\mathbb{C}(x)))$ and
$\displaystyle\phi_{3}=\phi(j_{*}(\mathcal{O}_{C})).$

######  Lemma 6.3.

For every $\sigma\in S_{12},$ we have that
$\phi_{1}<\phi_{0}<\phi_{1}+1\textnormal{ and }\phi_{3}<\phi_{2}<\phi_{3}+1.$

######  Proof 48.

It follows directly from the stability of
$i_{*}(\mathbb{C}(x)),i_{*}(\mathcal{O}_{C})$ and
$j_{*}(\mathbb{C}(x)),j_{*}(\mathcal{O}_{C}).$

Since every $\sigma\in S_{12}$ satisfies
$\phi_{1}<\phi_{0}<\phi_{1}+1\textnormal{ and }\phi_{3}<\phi_{2}<\phi_{3}+1,$
then by Remark 2.53 for $(m_{0},m_{1},\phi_{0},\phi_{1}),$ where $m_{0}=\lvert
Z(i_{*}(\mathbb{C}(x)))\rvert\textnormal{ and }m_{1}=\lvert
Z(i_{*}(\mathcal{O}_{C}))\rvert$ and for $(m_{2},m_{3},\phi_{2},\phi_{3}),$
where $m_{2}=\lvert Z(j_{*}(\mathbb{C}(x)))\rvert\textnormal{ and
}m_{3}=\lvert Z(j_{*}(\mathcal{O}_{C}))\rvert.$ Therefore, we obtain two
stability conditions $\sigma_{1}=(Z_{1},\mathcal{A}_{1})=(T_{1},f_{1})$ and
$\sigma_{2}=(Z_{2},\mathcal{A}_{2})=(T_{2},f_{2})$ in
$\operatorname{Stab}(C).$

We define the map

(6.2) $\displaystyle\pi\colon S_{12}$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{P}_{12}$ $\displaystyle\sigma$ $\displaystyle\mapsto$
$\displaystyle(\sigma_{1},\sigma_{2}).$

Note that
${\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}_{1}}}=Z_{1}\textnormal{
and }{\left.\kern-1.2ptZ\vphantom{\big{|}}\right|_{\mathcal{C}_{2}}}=Z_{2}.$

######  Remark 6.4.

The group $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ acts freely on
$\mathcal{P}_{12}$ by the definition of the action.

######  Lemma 6.5.

The map $\pi$ is well-defined, continuous, open and
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-equivariant.

######  Proof 49.

First note that $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ also acts
freely on $S_{12}.$ As $\pi$ is defined in terms of the slicing, we clearly
have a $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$-equivariant continuous
map from $S_{12}$ to
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})\times\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}).$

We will now show that $(\sigma_{1},\sigma_{2})\in\mathcal{P}_{12}.$ First, we
prove that $m-n\geq-1.$ Since $i_{*}(\mathbb{C}(x)),j_{*}(\mathbb{C}(x))$ are
stable and we have a non-zero morphism $i_{*}(\mathbb{C}(x))\rightarrow
j_{*}(\mathbb{C}(x))[1],$ it follows that $\phi_{0}-\phi_{2}<1.$

If $\phi_{0}>\phi_{2},$ then by Remark 4.10 we get that $l_{*}(\mathbb{C}(x))$
is stable. We show now that in this case $\det(M_{1}+M_{2})>0.$ By Proposition
4.21 and the analogous propositions for $\Theta_{23}$ and $\Theta_{31},$ we
obtain that there is $g\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ such
that by acting by $g$ we obtain a stability condition
$\sigma^{{}^{\prime}}=\sigma g$ such that
$\pi(\sigma^{{}^{\prime}})=(\sigma_{1}g,\sigma_{\mu}).$ Let
$\sigma^{{}^{\prime}}=\sigma_{1}g=(T^{{}^{\prime}},f^{{}^{\prime}})$ and
$M^{{}^{\prime}}={T^{{}^{\prime}}}^{-1}.$ By Lemma 4.30, we have
$\det(M^{{}^{\prime}}+I)>0.$ Note that $M^{{}^{\prime}}=M_{2}^{-1}M_{1},$
therefore

$0<\det(M_{2}^{-1}M_{1}+I)=\det(M_{2})\det(M_{1}+M_{2}).$

As $\phi_{3}<\phi_{2}<\phi_{3}+1,$ we obtain $\det(M_{2})>0.$ This implies
$\det(M_{1}+M_{2})>0.$ Moreover, as $i_{*}(\mathcal{O}_{C})$ and
$j_{*}(\mathcal{O}_{C})$ are $\sigma$-stable and there is a non-zero morphism
$i_{*}(\mathcal{O}_{C})\rightarrow j_{*}(\mathcal{O}_{C})[1],$ then we
immediately obtain that $\phi_{1}<\phi_{3}+1.$

Consequently, we obtain that $\pi$ is well defined.

Since $\pi^{\prime}\colon S_{12}\rightarrow\rm{GL}^{+}(2,\mathbb{R})^{2}$ is a
local homeomorphism, where $\pi^{\prime}$ maps a stability condition to its
stability function, the fact that $\pi$ is a local homeomorphism follows
almost directly from the fact that $p\circ\pi=\pi^{\prime},$ where
$p\colon\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})^{2}\rightarrow\rm{GL}^{+}(2,\mathbb{R})$
is the universal covering.

In order to prove that the map $\pi$ is in fact a homeomorphism, we study the
action of $\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ on $S_{12}.$ As
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ acts freely on $S_{12},$ we
define a section of the action

$\mathcal{V}_{12}=\\{\sigma\in
S_{12}\mid\sigma=\pi(\sigma_{1},\sigma_{2})\textnormal{ such that
}\sigma_{2}=\sigma_{\mu}\\}.$

######  Lemma 6.6.

If $\sigma\in\mathcal{V}_{12}$ and $\pi(\sigma)=(\sigma_{1},\sigma_{\mu})$
with $\sigma_{1}=(T,f)\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ then
$l_{*}(\mathbb{C}(x))$ is $\sigma$-stable if and only if $-1<f(0)<0.$

######  Proof 50.

If $l_{*}(\mathbb{C}(x))$ is $\sigma$-stable, since $i_{*}(\mathbb{C}(x))$ and
$j_{*}(\mathbb{C}(x))$ are also $\sigma$-stable, we obtain that
$\phi_{2}<\phi_{4}<\phi_{0}<\phi_{2}+1$ and therefore that $n=-1$ and
$-1<f(0)<0.$ If $l_{*}(\mathbb{C}(x))$ is not $\sigma$-stable, then by Remark
4.7 we have $\phi_{0}-\phi_{2}\leq 0,$ which implies that $f(0)\geq 0$ and we
obtain a contradiction.

######  Remark 6.7.

If $\sigma\in\mathcal{V}_{12}$ and $\pi(\sigma)=(\sigma_{1},\sigma_{\mu})$ and
$0\leq f(0),$ then as $l_{*}(\mathbb{C}(x))$ is not $\sigma$-stable, by
Theorem 4.13 we have $\sigma\in\Theta_{12}.$

The image of $\mathcal{V}_{12}$ under $\pi$ is contained in

$\mathcal{L}_{12}=\\{(\sigma,\sigma_{\mu})\in\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})^{2}\mid
f(0)>-1\textnormal{, }3/2>f^{-1}(1/2)\textnormal{ and if}f(0)<0\textnormal{
then }\det(M+I)>0\\}.$

Abusing the notation we see $\mathcal{L}_{12}$ as a subset of
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}).$

######  Lemma 6.8.

The subset
$\mathcal{L}_{12}\subseteq\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R})$ is
open and connected.

######  Proof 51.

By definition $\mathcal{L}_{12}$ is clearly an open subset of
$\widetilde{\textnormal{GL}}^{+}(2,\mathbb{R}).$

We define $Y\subseteq\mathbb{R}^{4}$ as follows: We say that
$(m_{0},m_{1},\phi_{0},\phi_{1})\in Y$ if $m_{i}>0,\textnormal{
}\phi_{0}<2\textnormal{, }$$\phi_{1}<\frac{3}{2}\textnormal{,
}\phi_{1}<\phi_{0}<\phi_{1}+1$ and if $1\leq\phi_{0}<2\textnormal{ and
}0<\phi_{1}<\frac{3}{2}\textnormal{, then
}\delta(m_{0},m_{1},\phi_{0},\phi_{1})>-1,$ where

$\displaystyle\delta\colon\mathbb{R}_{>0}\times\mathbb{R}_{>0}\times(1,2)\times(0,\frac{3}{2})$
$\displaystyle\rightarrow$ $\displaystyle\mathbb{R}$
$\displaystyle(m_{0},m_{1},\phi_{0},\phi_{1})$ $\displaystyle\mapsto$
$\displaystyle
m_{0}m_{1}\sin((\phi_{0}-\phi_{1})\pi)-m_{0}\cos(\phi_{0}\pi)+m_{1}\sin(\phi_{1}\pi).$

Note that $Y$ is connected, because although $f$ is defined in terms of
trigonometric functions, it is restricted to intervals where it behaves well.
It is easy to see that the map

(6.3) $\displaystyle\rho\colon\mathcal{L}_{12}$ $\displaystyle\rightarrow$
$\displaystyle Y$ $\displaystyle(T,f)$ $\displaystyle\mapsto$
$\displaystyle(m_{0},m_{1},f^{-1}(1),f^{-1}(\frac{1}{2}))$

where $m_{0}=|A+Di|,m_{1}=|B+Ci|,$ is a homeomorphism. It follows that
$\mathcal{L}_{12}$ is connected.

######  Proposition 6.9.

The map

$\displaystyle\pi\colon$ $\displaystyle\mathcal{V}_{12}$
$\displaystyle\rightarrow\mathcal{L}_{12}$ $\displaystyle\sigma$
$\displaystyle\mapsto\sigma_{1},$

is a homeomorphism.

######  Proof 52.

First, we prove that $\pi$ is injective. Let $\sigma=(Z,\mathcal{A})$,
$\tau=(W,\mathcal{B})\in\mathcal{V}_{12},$ such that
$\pi(\sigma)=\pi(\tau)=\sigma_{1}.$ If $0\leq f(0),$ by Remark 6.7, we obtain
$\sigma,\tau\in\Theta_{12}.$ By Lemma 4.26, it follows that
$\sigma=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})=\tau.$

If $0>f(0)>-1,$ by Lemma 6.6, we have that $l_{*}(\mathbb{C}(x))$ is stable.
In Lemma 4.30 we described this type of stability conditions and its hearts.
As by definition $Z=W$ and $d(\mathcal{P},\mathcal{Q})<1,$ where $\mathcal{P}$
and $\mathcal{Q}$ are the slicing of $\sigma$ and $\tau$ respectively, then
$\sigma=\tau$.

Thus, by Lemma 6.5 we already have a homeomorphism onto the image of
$\mathcal{V}_{12}.$ We now prove that it is in fact onto. By Lemma 6.1 and
Lemma 6.5 the image of $\mathcal{V}_{12}$ is open. Since $\mathcal{L}_{12}$ is
also connected, it is enough to prove that $\pi(\mathcal{V}_{12})$ is closed.
Moreover, it contains a dense subset as the image of the discrete stability
conditions constructed in Section LABEL:CSCG and Lemma 3.48. We prove it by
contradiction. Assume that $\pi(\mathcal{V}_{12})$ is not close and let us
take a $\tau_{1}=(Z,\mathcal{A})$ in the boundary of $\pi(\mathcal{V}_{12})$
which does not belong to the image. Note that there is
$\tau^{{}^{\prime}}=(Z^{{}^{\prime}},\mathcal{A}^{{}^{\prime}})=\pi(\sigma^{{}^{\prime}})\in\pi(\mathcal{V}_{12}),$
where $\sigma^{\prime}=(W^{\prime},\mathcal{B^{\prime}}),$ sufficiently close
to $\tau_{1}$ such for $W=Z(r_{1},d_{1})+Z_{\mu}(r_{2},d_{2}),$ we have that
$Q_{W^{\prime}}$ restricted to $\operatorname{Ker}{W}$ is negative definite,
where $Q_{W^{\prime}}$ is the quadratic form given by the support property
satisfied by $\sigma^{\prime}.$

Consider the open subset of $\operatorname{Hom}(\mathbb{Z}^{4},\mathbb{C}),$
consisting in homomorphisms whose kernel is negative definite with respect to
$Q_{W^{\prime}}$ and let $U$ be the connected component containing
$W^{\prime}.$ Then as in [5, Proposition A.5], there is a continuos function
$C\colon U\rightarrow\mathbb{R}_{>0}$ such that $C(Y)\in\mathbb{R}_{>0}$
satisfies that $||v||C(Y)<|Y(v)|$ for $v\in\mathbb{R}^{4}$ with
$Q_{W^{\prime}}(v)\geq 0,$ an appropiate norm in $\mathbb{R}^{4}$ and $|-|$
the Euclidean norm in $\mathbb{C}.$ Then there is $0<\epsilon<\frac{1}{8},$
such that $|W-W^{{}^{\prime}}|_{\infty}\leq\sin(\pi\epsilon)C(W^{\prime}).$ As
a consequence we have that
$|W-W^{{}^{\prime}}|_{\infty}<\sin(\epsilon\pi)\frac{|W^{\prime}(E)|}{||[E]||},$
which implies $||W-W^{\prime}||_{\sigma^{\prime}}<\sin(\pi\epsilon).$

By Bridgeland’s deformation Theorem 2.43, there is a stability condition
$\sigma=(W,\mathcal{B})$ in the neighbourhood of $\sigma^{{}^{\prime}}.$ It is
possible to choose $\tau^{{}^{\prime}}$ appropriately, such that $\sigma$ is
in a desired wall. Now we assume the object $j_{*}(\mathbb{C}(x))$ to be
$\sigma$-semistable but not stable and
$j_{*}(\mathbb{C}(x))\in\mathcal{P}_{\sigma}(1).$ By Remark 4.10 the Jordan-
Hölder filtration is given by $i_{*}(\mathbb{C}(x))[-1]\rightarrow
j_{*}(\mathbb{C}(x))\rightarrow l_{*}(\mathbb{C}(x)),$ all with the same
phase. Therefore, we obtain
$i_{*}(\mathbb{C}(x))[-1]\in\mathcal{P}_{\sigma}(1),$ which implies that
${\left.\kern-1.2ptW\vphantom{\big{|}}\right|_{\mathcal{C}_{1}}}=Z=Ad+Br+iCr.$
It follows that $\tau_{1}$ is a stability condition given by
$(Z,\operatorname{Coh}(C)[n]),$ for $n\in\mathbb{Z}.$ By the definition of
$\mathcal{L}_{12}$ we have that $f(0)=n\geq 0.$ As a consequence, the
stability conditions $\tau_{1}$ and $\sigma_{\mu}$ satisfy the CP-gluing
conditions. Moreover
$\sigma_{1}=\operatorname{gl}_{12}(\tau_{1},\sigma_{\mu})$ is a discrete
stability condition, by Corollary 3.36 and Lemma 5.20 a Bridgeland stability
condition in $\mathcal{V}_{12}$, satisfying $\pi(\sigma_{1})=\tau_{1}.$ We
obtain a contradiction. The argument goes along the same lines for the other
walls. We finally obtain $\pi(\mathcal{V}_{12})=\mathcal{L}_{12}.$

######  Corollary 6.10.

The map

$\displaystyle\pi\colon$ $\displaystyle S_{12}$
$\displaystyle\rightarrow\mathcal{P}_{12}$ $\displaystyle\sigma$
$\displaystyle\mapsto(\sigma_{1},\sigma_{2})$

is a homeomorphism.

We can finally prove the Harder–Narasimhan property for non-discrete stability
conditions.

######  Proposition 6.11.

The pairs $\sigma=(Z,\mathcal{A})$ constructed in Corollary 3.36 with the
semiorthogonal decomposition $\langle\mathcal{C}_{1},\mathcal{C}_{2}\rangle$
and the pairs $\sigma=(Z_{r},\mathcal{A}_{r})$ given in Lemma 3.47 with
$f^{-1}(\frac{1}{2})<\frac{3}{2}$ are Bridgeland stability conditions.

######  Proof 53.

Whenever
$\sigma_{1}=(T_{1},f_{1})=(Z_{1},\mathcal{A}_{1})\in\operatorname{Stab}(C)$
and $\sigma_{2}$ are discrete we have already shown that the data gives us a
Bridgeland stability condition. It is enough to show this whenever
$\sigma_{2}=\sigma_{\mu}.$ Let
$\sigma=\operatorname{gl}_{12}(\sigma_{1},\sigma_{\mu})$ be a CP-glued pair.
As $f_{1}(0)\geq 0,$ we get $(\sigma_{1},\sigma_{\mu})\in\mathcal{L}_{12}.$
Therefore, there is a stability condition $\tau\in\mathcal{V}_{12}$ such that
$\pi(\tau)=(\sigma_{1},\sigma_{2}).$ By Remark 6.7, we obtain
$\tau\in\Theta_{12}$ and by Lemma 4.26, we have that
$\tau=\operatorname{gl}_{12}(\sigma_{1},\sigma_{2})=(Z,\mathcal{A}).$ As a
consequence, the pair $(Z,\mathcal{A})$ gives a Bridgeland stability
condition.

Let $\sigma=(Z_{r},\mathcal{A}_{r}).$ We consider $\sigma_{1}$ as in Lemma
3.47 with $f^{-1}(\frac{1}{2})<\frac{3}{2}.$ We get that $-1<f_{1}(0)<0$ and
by hypothesis $\det(M_{1}+I)>0,$ we have that
$(\sigma_{1},\sigma_{\mu})\in\mathcal{L}_{12}.$ As a consequence, there is a
stability condition $\tau\in\mathcal{V}_{12}$ such that
$\pi(\tau)=(\sigma_{1},\sigma_{2}).$ By Lemma 6.6, we obtain that
$l_{*}(\mathbb{C}(x))$ is $\tau$-stable. Therefore by Lemma 4.30, we have that
$\tau$ is given precisely by the construction in Proposition 3.47. Then
$\tau=(Z_{r},\mathcal{A}_{r})$ and as a consequence, it is a Bridgeland
stability condition.

######  Remark 6.12.

If $\sigma=(Z_{r},\mathcal{A}_{r})$ is a pre-stability condition given in
Lemma 3.47 with $f^{-1}(\frac{1}{2})\geq\frac{3}{2},$ then either
$i_{*}(\mathcal{O}_{C})$ is not stable or $j_{*}(\mathcal{O}_{C})$ is not
stable. Then $\sigma$ is in $S_{23}$ or in $S_{31}.$ Precisely, by Lemma 4.16
if $\phi_{5}>\frac{3}{2}$ then $j_{*}(\mathcal{O}_{C})$ and
$l_{*}(\mathcal{O}_{C})$ are $\sigma$-stable, and if $\phi_{5}<1/2$ then