$\nu_{\alpha,\beta}$-semistability of $B$.

Lemma 5.15 implies that $\operatorname{ch}_{0}(P)=0$. If
$\operatorname{ch}_{1}(P)\neq 0$, then

$\lim_{\beta\to\infty}\dfrac{1}{\beta}(\nu_{\overline{\alpha},\beta}(B)-\nu_{\overline{\alpha},\beta}(P))=\dfrac{1}{2},$

also contradicting the asymptotic $\nu_{\alpha,\beta}$-semistability of $B$.

Any $U$ is a subsheaf of $P$ has dimension less than 1, so
$\nu_{\alpha,\beta}(U)=+\infty$ and hence it cannot lift to a sub-object of
$B$. By Proposition 2.15, we conclude that $S:=B^{\vee}[-1]$ is a torsion free
sheaf; note that ${\mathcal{E}}{\it
xt}^{3}(S^{**}/S,{\mathcal{O}}_{X})\simeq{\mathcal{E}}{\it
xt}^{2}(S,{\mathcal{O}}_{X})={\mathcal{H}}^{1}(B)=0$, so $S^{**}/S$ cannot
have a 0-dimensional subsheaf.

Let $G$ be a quotient sheaf of $S$, hence $G^{*}$ is a subsheaf of
$S^{*}={\mathcal{H}}^{-1}(B)$, and $G^{*}[1]\hookrightarrow B$ is a sub-object
within $\mathcal{B}^{\beta}(X)$ for every $\beta>\mu^{+}(G^{*})$. Note that
$\delta_{10}(G^{*}[1],B)=\delta_{10}(S,G)$, so

$\lim_{\beta\to\infty}(\nu_{\overline{\alpha},\beta}(G^{*}[1])-\nu_{\overline{\alpha},\beta}(B))=\dfrac{1}{2}\dfrac{\delta_{10}(S,G)}{\operatorname{ch}_{0}(G)\operatorname{ch}_{0}(S)}\leq
0$

by asymptotic $\nu_{\alpha,\beta}$-semistability, so $S$ is $\mu$-semistable.

Next, note that $\delta_{20}(G^{*}[1],B)=\delta_{20}(G,S)$, thus if
$\delta_{10}(S,G)=0$, then

$\lim_{\beta\to\infty}(\beta)(\nu_{\overline{\alpha},\beta}(G^{*}[1])-\nu_{\overline{\alpha},\beta}(B))=-\dfrac{\delta_{20}(G,S)}{\operatorname{ch}_{0}(G)\operatorname{ch}_{0}(S)}\leq
0,$

implying that $\delta_{20}(S,G)\leq 0$, meaning that $S$ is $\mu_{\leq
2}$-semistable. ∎

Finally, we provide the converse of the previous lemma, thus concluding the
proof of Theorem 5.2. For $E\in\mathcal{C}oh(X)$, note Lemma 5.15 implies that
$E^{\vee}[-1]\in\mathcal{B}^{\beta}$ for $\beta\gg 0$ if and only if $E$ has
no subsheaf of dimension at most 1, and the cokernel of the canonical morphism
$E\to E^{**}$ has pure dimension 1.

######  Lemma 5.17.

If $S$ is a $\mu_{\leq 2}$-semistable sheaf such that $S^{**}/S$ is either
empty or has pure dimension 1, then there is $\beta_{0}>0$ such that
$S^{\vee}[-1]$ is $\nu_{\overline{\alpha},\beta}$-semistable for all
$\beta>\beta_{0}$.

###### Proof.

The observation in the previous paragraph implies that
$B:=S^{\vee}[-1]\in\mathcal{B}^{\beta}$ for every $\beta\geq-\mu(S)$.

Let $F\hookrightarrow B$ be a sub-object within $\mathcal{B}^{\beta}$ for
$\beta>-\mu(S)$; note that $\operatorname{ch}_{0}({\mathcal{H}}^{0}(F))=0$ by
Lemma 5.15. Assume first that
$\operatorname{ch}_{0}(F)=\operatorname{ch}_{0}({\mathcal{H}}^{-1}(F))\neq 0$.
It follows that

$\mu(F)=\mu({\mathcal{H}}^{-1}(F))-\dfrac{\operatorname{ch}_{1}({\mathcal{H}}^{0}(F))}{\operatorname{ch}_{1}({\mathcal{H}}^{-1}(F))}\leq\mu({\mathcal{H}}^{-1}(F))\leq\mu(S^{*})=\mu(B)$

since $S^{*}$ is $\mu$-semistable and ${\mathcal{H}}^{-1}(F)$ is a subsheaf of
$S^{*}$. We then have

$\lim_{\beta\to\infty}\left(\nu_{\overline{\alpha},\beta}(F)-\nu_{\overline{\alpha},\beta}(B)\right)=\dfrac{1}{2}\left(\mu(F)-\mu(B)\right)\leq
0.$

If $\mu(F)=\mu(B)$, then $\mu({\mathcal{H}}^{-1}(F))=\mu(S^{*})$ and
$\operatorname{ch}_{1}({\mathcal{H}}^{0}(F))=0$. Denoting $G:=B/F$ the
corresponding quotient in $\mathcal{B}^{\beta}$, we have the following exact
sequence in $\mathcal{C}oh(X)$:

(36) $0\to{\mathcal{H}}^{-1}(F)\to
S^{*}\to{\mathcal{H}}^{-1}(G)\stackrel{{\scriptstyle
f}}{{\to}}{\mathcal{H}}^{0}(F)\to{\mathcal{E}}{\it
xt}^{1}(S,{\mathcal{O}}_{X})\to{\mathcal{H}}^{0}(G)\to 0;$

Clearly, $\ker f$ is torsion free, and this implies that
${\mathcal{H}}^{-1}(F)$ is reflexive. Let
$P\hookrightarrow{\mathcal{H}}^{0}(F)$ be a monomorphism of sheaves that lifts
to $F$, and let $U$ be the image of the composite morphism of sheaves
$P\hookrightarrow{\mathcal{H}}^{0}(F)\to{\mathcal{E}}{\it
xt}^{1}(S,{\mathcal{O}}_{X})$. Since
$U\in\mathcal{T}_{\beta}\subset\mathcal{B}^{\beta}$ for every $\beta$, $U$
coincides with the image of the composite monomorphism $P\hookrightarrow
F\hookrightarrow B=S^{\vee}[-1]$ in $\mathcal{B}^{\beta}$, so Proposition 2.15
implies that $U=0$, so in fact $P=0$. We therefore conclude, again by
Proposition 2.15, that $\overline{F}:=F^{\vee}[-1]$ is a torsion free sheaf.
Dualizing and shifting the triangle

$F\to B\to G\to F[1]$

in $D^{\rm b}(X)$, we obtain the triangle

$\overline{G}\to S\to\overline{F}\to G^{\vee}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\overline{G}:=G^{\vee}[-1]$

which yields the following exact sequence in $\mathcal{C}oh(X)$

$0\to{\mathcal{H}}^{0}(\overline{G})\to
S\to\overline{F}\to{\mathcal{H}}^{1}(\overline{G})\to 0.$

Note that $\operatorname{ch}_{k}({\mathcal{H}}^{1}(\overline{G}))=0$ for
$k=0,1$; it follows that

$\delta_{20}(B,G)=\delta_{20}(S,\overline{G})=\delta_{20}(S,{\mathcal{H}}^{0}(\overline{G}))+\operatorname{ch}_{0}(S)\operatorname{ch}_{2}({\mathcal{H}}^{1}(\overline{G}))\geq
0.$

We then obtain

$\lim_{\beta\to\infty}\beta\left(\nu_{\overline{\alpha},\beta}(B)-\nu_{\overline{\alpha},\beta}(G)\right)=-\dfrac{\delta_{20}(B,G)}{\operatorname{ch}_{0}(B)\operatorname{ch}_{0}(G)}\leq
0$

If the equality holds, then both $\operatorname{ch}_{\leq 2}(F)$ and
$\operatorname{ch}_{\leq 2}(G)$ are multiples of $\operatorname{ch}_{\leq
2}(B)$, meaning that
$\nu_{\alpha,\beta}(F)=\nu_{\alpha,\beta}(B)=\nu_{\alpha,\beta}(G)$ for every
$(\alpha,\beta)$. In any case, we conclude that there is $\beta_{0}>0$ such
that $\nu_{\overline{\alpha},\beta}(F)\leq\nu_{\overline{\alpha},\beta}(B)$
for $\beta>\beta_{0}$.

Next, assume that
$\operatorname{ch}_{0}(F)=\operatorname{ch}_{0}({\mathcal{H}}^{-1}(F))=0$, so
${\mathcal{H}}^{-1}(F)=0$ and $F\in\mathcal{T}_{\beta}$. If
$\operatorname{ch}_{1}(F)\neq 0$, then

$\lim_{\beta\to\infty}\dfrac{1}{\beta}(\nu_{\overline{\alpha},\beta}(B)-\nu_{\overline{\alpha},\beta}(F))=\dfrac{1}{2},$

so $\nu_{\overline{\alpha},\beta}(B)>\nu_{\overline{\alpha},\beta}(F)$ for
$\beta\gg 0$. Therefore, it is enough to consider the case when $\dim F\leq
1$.

The sequence in display (36) simplifies to

$0\to S^{*}\to{\mathcal{H}}^{-1}(G)\stackrel{{\scriptstyle
f}}{{\to}}{\mathcal{H}}^{0}(F)\to{\mathcal{E}}{\it
xt}^{1}(S,{\mathcal{O}}_{X})\to{\mathcal{H}}^{0}(G)\to 0.$

Dualizing the sequence $0\to S^{*}\to{\mathcal{H}}^{-1}(G)\to\ker f\to 0$ we
get that $S^{**}\simeq{\mathcal{H}}^{-1}(G)^{*}$, and therefore $\ker f=0$.
This means that $F$ is a subsheaf of ${\mathcal{E}}{\it
xt}^{1}(S,{\mathcal{O}}_{X})$ that lifts to $B=S^{\vee}[-1]$, contradicting
Proposition 2.15. ∎

##  6\. The Differential Geometry of Surface Walls

######  Definition 6.1.

Let $u$ and $v$ be real numerical Chern characters such that $v$ satisfies the
Bogomolov inequality. We define the surface
$\Sigma_{u,v}\subset\mathbb{R}^{+}\times\mathbb{R}\times\mathbb{R}^{+}$ to be
the vanishing locus of

$f_{u,v}(\alpha,\beta,s)=\Delta_{32}(\alpha,\beta)-\alpha^{2}(s-1/3)\Delta_{12}(\alpha,\beta),$

the numerator of the difference of slopes
$\lambda_{\alpha,\beta,s}(u)-\lambda_{\alpha,\beta,s}(v)$ now regarded as a
function of all three parameters $(\alpha,\beta,s)$. Note that

(37) $\Sigma_{u,v}=\Sigma_{v,u}=\Sigma_{\phi v+\psi u,v},$

for any real $\phi$ and $\psi\neq 0$.

In addition, we will denote by $\Gamma_{v}$ (without the parameter $s$ in the
subscript) the surface
$\\{\tau_{v,s}(\alpha,\beta)=0\\}\subset\mathbb{R}\times\mathbb{R}^{+}$. In
this notation, $\Gamma_{v,s_{0}}=\Gamma_{v}\cap\\{s=s_{0}\\}$ and
$\Upsilon_{u,v,s_{0}}=\Sigma_{u,v}\cap\\{s=s_{0}\\}$, for any
$s_{0}\in\mathbb{R}^{+}$.

Our aim in this section is to explore some of the differential geometric
properties of $\Sigma_{u,v}$ with a view to understanding finiteness
properties of $\lambda$-walls. We will assume that $v$ is a fixed real
numerical Chern character. Note that, by (37) we can assume that
$u_{0}=v_{0}$. It turns out to be best to consider the two cases
$u_{0}=0=v_{0}$ and $u_{0}\neq 0\neq v_{0}$. The former is dealt with in
Remark 6.5.

Generically, $\Sigma_{u,v}$ can have components of dimensions $0$, $1$ or $2$.
In fact, each of these do arise at least as numerical surface walls. For
example, for $v=(3,-2,-1/2,1)$ and $u=(0,1,-1/2,0)$

$f_{u.v}(0,\beta,s)=(\beta+1)^{2}(\beta^{2}+2)/4$

and $f_{u,v}(\alpha,\beta,s)\neq 0$ whenever $\alpha\neq 0$. At special points
where the curves $\Gamma_{v,s}$ and $\Theta_{v}$ intersect, $\Sigma_{u,v}$ is
guaranteed to be locally a surface.

######  Lemma 6.2.

At the points of intersection $\Gamma_{u,s}\cap\Gamma_{v,s}$,
$\Theta_{u}\cap\Theta_{v}$ and $\Gamma_{v,s}\cap\Theta_{v}$, $\Sigma_{u,v}$ is
2-dimensional.

In other words, a surface wall $\Sigma_{u,v}$ does have 2-dimensional
components whenever it is not empty.

###### Proof.

Note that each pair of curves divides a small ball around their intersection
into 4 regions otherwise the (algebraic) curves must coincide. They cannot
coincide except possibly for the Theta curves but then
$\Sigma_{u,v}=\Theta_{u}\times\mathbb{R}$. Then for a small arc around the
intersection point in two of the regions, the function
$f_{u,v}(\alpha,\beta,s)$ is positive on one curve and negative on the other,
and so must vanish at some point in the region for each arc sufficiently close
to the intersection point. Combining with Lemma 4.11 we see that
$\Sigma_{u,v}$ is 2-dimensional in a neighbourhood of the point. ∎

We now look more carefully at the differential geometry of the surface wall
$\Sigma_{u,v}$. The normal vector is given by the gradient of
$f(\alpha,\beta,s)$.

######  Lemma 6.3.

$\displaystyle\partial_{\alpha}f$
$\displaystyle=\alpha\bigl{(}1+2(s-1/3)\bigr{)}\Delta_{21}-\alpha\Delta_{30}+\alpha^{3}(s-1/3)\Delta_{10}.$
$\displaystyle\partial_{\beta}f$
$\displaystyle=-\Delta_{31}-\alpha^{2}(s-1/3)\Delta_{20}.$
$\displaystyle\partial_{s}f$ $\displaystyle=\alpha^{2}\Delta_{21}.$
$\displaystyle\partial^{2}_{\alpha}f$
$\displaystyle=(1+2(s-1/3))\Delta_{21}-\Delta_{30}+\alpha^{2}(5s+1/3)\Delta_{10}.$
$\displaystyle Hf$
$\displaystyle=\begin{pmatrix}\partial^{2}_{\alpha}f&2\alpha(s-1/3)\Delta_{02}&2\alpha\Delta_{21}-\alpha^{3}\Delta_{01}.\\\
2\alpha(s-1/3)\Delta_{02}&\Delta_{21}+\Delta_{30}+\alpha^{2}(s-1/3)\Delta_{10}&-\alpha^{2}\Delta_{20}.\\\
2\alpha\Delta_{21}-\alpha^{3}\Delta_{01}&-\alpha^{2}\Delta_{20}&0\end{pmatrix}.$

######  Proposition 6.4.

For any $s_{1}>s_{0}>0$,
$\Sigma_{u,v}\cap\mathbb{R}\times\mathbb{R}\times\\{s_{0}\leq s\leq s_{1}\\}$
is compact if and only if $\Delta_{01}\neq 0$.

###### Proof.

This follows from Lemma 4.8 and its converse which follows because the leading
homogeneous terms of $f(\alpha,\beta,s)$ (up to a constant) are

$\bigl{(}6s\alpha^{2}(\alpha^{2}+\beta^{2})+(\beta^{2}-\alpha^{2})^{2}\bigr{)}\Delta_{01}$

which vanishes for $s>0$ only if either $\alpha=\beta=0$, or $\Delta_{01}=0$.
∎

Note that when $s=0$ the surface is unbounded along $\alpha=\pm\beta$.

######  Remark 6.5.

In the case where $v_{0}$ and $u_{0}$ are non-zero, if $u$ gives rise to a
$\nu$-wall with respect to $v$ then $\Delta_{01}\neq 0$. This is then
$\Delta_{21}(\alpha,\beta)=0$ has a 1-dimensional locus. So if $\Delta_{01}=0$
(everywhere) then $\Delta_{20}=0$ along the $\nu$-wall and it follows that
$(u_{0},u_{1},u_{2})\propto(v_{0},v_{1},v_{2})$ which then cannot have a
$\nu$-wall as $\Delta_{21}=0$ everywhere. In fact, if $\Delta_{01}=0$ then if
$\Delta_{20}(\alpha,\beta)=0$ for some $(\alpha,\beta)$ it must vanish
identically and again $(u_{0},u_{1},u_{2})\propto(v_{0},v_{1},v_{2})$. But
then $\Delta_{21}=0$ identically. On a $\lambda-$wall we then have
$\Delta_{32}=0$ and so $u\propto v$. Hence, $\Delta_{02}\neq 0$.

If $v_{0}=0=u_{0}$ then $\Delta_{i0}=0$ identically for all $i$. Then there
are no $\nu$-walls. The numerical $\lambda$-walls are nested ellipses much as
$\nu$-walls are for the truncated Chern characters. It follows that
$\Sigma_{u,v}$ is always regular and horizontal exactly on $\Gamma_{v,s}$. The
variation in $s$ is just a vertical scaling by $\sqrt{s+1/3}$.

It is interesting to consider the regularity of numerical walls and we will
use this extensively in our analysis of the asymptotics. The regularity of a
general wall for arbitrary $s$ is complicated and hard to describe but the
situation is simpler if we consider the whole surface $\Sigma_{u,v}$ and also
for the case $s=1/3$ which we describe first.

######  Proposition 6.6.

When $s=1/3$ a numerical wall $\Upsilon_{u,v,1/3}$ is regular everywhere in
the upper half plane except where it intersects $\Gamma_{u,1/3}$ and its
numerical $\nu$-wall $\Xi_{u,v}$.

###### Proof.

At a non-regular point $p$ on $\Upsilon_{u,v,1/3}$ we have
$\Delta_{32}=0=\Delta_{31}$ and $\Delta_{30}=\Delta_{21}$. If
$p\not\in\Gamma_{v,1/3}$ then Lemma 2.7(3) implies $\Delta_{12}=0$ and so
$\Delta_{30}=0$. But then Lemma 2.7(4) implies $u\propto v$. So
$p\in\Gamma_{v,1/3}\cap\Gamma_{u,1/3}$. But then $\Delta_{30}=0$ and so
$\Delta_{21}=0$. But then $p\in\Xi_{u,v}$. Conversely, if
$p\in\Gamma_{v,1/3}\cap\Xi_{u,v}\cap\Upsilon_{u,v,1/3}$ then
$\partial_{\alpha}f=0=\partial_{\beta}f$ and so $p$ is not a regular point. ∎

Looking at the second derivatives we see that the local model for
$\Upsilon_{u,v,q/3}$ at its singular point is
$(\alpha-\alpha_{0})^{2}+\text{higher order terms}=0$ and so the singular
point is a cusp. We see this more generally in case (2) in the Theorem.

######  Theorem 6.7.

Suppose $u$ and $v$ are real numerical Chern characters with $v_{0}\neq 0$.
Any 2-dimensional component of the surface wall $\Sigma_{u,v}$ is regular
everywhere except in one of the following situations:

  1. (1)

it intersects a $\nu$-wall away from $\Gamma_{v}$ at $\alpha=0$, in which case
it is locally $\alpha^{2}-\beta^{2}$;

  2. (2)

it intersects a $\nu$-wall and $\Gamma_{v}$ at $s=1/3$ for $\alpha>0$ in which
case it is locally
$(\alpha-\alpha_{0})^{2}+(\alpha-\alpha_{0})(s-1/3)+\dfrac{\Delta_{02}(\alpha_{0},\beta_{0})}{\Delta_{01}(\alpha_{0},\beta_{0})}(\beta-\beta_{0})(s-1/3)=0$
up to scale;

  3. (3)

it intersects a $\nu$-wall, $\Gamma_{v}$, $\alpha=0$ and $s=1/3$ in which case
the surface is smooth (locally given by $(\beta-\beta_{0})^{3}=0$), or

  4. (4)

$u$ and $v$ are special vectors satisfying $\mu(v)=\mu(u)$,
$\Delta_{03}=\Delta_{21}$, $\Delta_{20}\neq 0$ and
$\operatorname{ch}^{\alpha,\mu(v)}_{3}(v)=0=\operatorname{ch}^{\alpha,\mu(u)}_{3}(u)$.
Then $\Sigma_{u,v}$ is regular except along the line
$M_{v}\times\\{s=1/3\\}\cup\\{\alpha=0,\ s<1/3\\}$ and it is locally
$(\beta-\beta_{0})\bigl{(}(s_{0}-1/3)(\alpha-\alpha_{0})+(\beta-\beta_{0})^{2}\bigr{)}$
at $(\alpha_{0},\mu(v),1/3)$ and at $(0,\beta_{0},s_{0})$.

Note that in (1) we also allow the degenerate case where the $\nu$-wall has
zero radius. An example of this can be seen for $v=(2,0,-1,0)$ illustrated in
Figure 11 in Section 8.

###### Proof.

Suppose $(\alpha,\beta,s)$ is a non-regular point of $\Sigma_{u,v}$. Then
$\nabla f_{u,v}(\alpha,\beta)=0$. From $\partial_{s}f_{u,v}=0$ we have either
$\alpha=0$ or $\Delta_{12}(\alpha,\beta)=0$. We will treat these two cases
separately.

First suppose $\alpha\neq 0$. Then $\Delta_{12}=0$. From $f_{u,v}=0$, we also
have $\Delta_{32}=0$. From $\partial_{\alpha}f_{u,v}=0$ we have
$\Delta_{30}=\alpha^{2}(s-1/3)\Delta_{10}$ and from
$\partial_{\beta}f_{u,v}=0$ we have
$\Delta_{31}=-\alpha^{2}(s-1/3)\Delta_{20}$. Assume that
$\operatorname{ch}_{1}^{\beta}(v)\neq 0$ and $s\neq 1/3$. Then Lemma 2.7 item
(3) implies that $\Delta_{31}=0$ and then $\Delta_{20}=0$ so that also
$\Delta_{10}=0$. Since $v_{0}\neq 0$, Lemma 2.7 item (4) implies $u\propto v$
which is a contradiction. So we must have
$\operatorname{ch}_{1}^{\beta}(v)=0=\operatorname{ch}_{1}^{\beta}(u)$ or
$s=1/3$. In the case $s\neq 1/3$, it follows that $\Delta_{01}=0$. From
$\partial_{\beta}f_{u,v}=0$ we have $\Delta_{02}=0$ but this contradicts
$\Delta_{03}=0$. So we must have have $s=1/3$ and either (a)
$\Delta_{01}=0=\Delta_{30}$ along $\operatorname{ch}^{\beta}_{1}(v)=0$ or (b)
the $\lambda$ curve passes through both a $\nu$-wall and $\Gamma$.

We consider case (a). From $\Delta_{31}=0$ we then have that $u\propto v$
unless
$\operatorname{ch}_{3}^{\alpha,\beta}(u)=0=\operatorname{ch}_{3}^{\alpha,\beta}(v)$
along $M_{v}$. Then we have that the conditions are equivalent to

(38) $\displaystyle\Delta_{01}=0,$ (39) $\displaystyle
v_{3}v_{0}^{2}-v_{1}v_{2}v_{3}+v_{1}^{3}/3=0,$ (40)
$\displaystyle\Delta_{03}=\Delta_{12},\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm and}$ (41) $\displaystyle\Delta_{02}\neq 0.$

The first equality for $u$ also follows (from the second). Note that in this
situation $\Sigma_{u,v}$ is singular all along $M_{v}$ and so also also at
$\alpha=0$. Along $M_{v}$ the Hessian of $f$ vanishes. The only third
derivatives to be non-zero (up to symmetry) are
$\partial_{\beta}^{3}f_{u,v}=2\Delta_{02}$ and
$\partial_{\alpha}\partial_{\beta}\partial_{s}f_{u,v}=2\alpha\Delta_{02}$. So
the local model for $f_{u,v}$ is

$2\Delta_{02}(\beta-\mu(v))\bigl{(}(\beta-\mu)^{2}+(\alpha-\alpha_{0})(s-1/3)\bigr{)}.$

This is a triple zero at $\alpha=0$ and looks like $y=0$ union $y^{2}=xz$ for
$\alpha\neq 0$. In fact, since $f_{u,v}(\alpha,v_{1}/v_{0})=0$ we have that
$(\beta-v_{1}/v_{0})$ is a linear factor of $f_{u,v}$.

Now suppose $\alpha=0$. Then $\partial_{\beta}f_{u,v}=0$ implies
$\Delta_{31}(\alpha,\beta)=0$. But from $f_{u,v}=0$ we have
$\Delta_{32}(\alpha,\beta)=0$. If the point is not on $\Gamma$ then
$\Delta_{21}(\alpha,\beta)=0$ and so $(\alpha,\beta)$ is on a $\nu$-wall. Then
the Hessian at this point is diagonal
$\operatorname{diag}(-\Delta_{30}(0,\beta),\Delta_{30}(0,\beta),0)$. Then
$f_{u,v}$ is locally of the form
$\Delta_{03}(\alpha^{2}-(\beta-\beta_{0})^{2})$ to lowest order as required.
Conversely, such a point (where a numerical $\lambda$-wall intersects its
associated $\nu$-wall away from $\Gamma_{v}$ on $\alpha=0$) is always singular
of this form.

Alternatively, if the point lies on $\Gamma_{v,s}\cap\Gamma_{u,s}$ and
$\alpha=0$ then the Hessian is
$\operatorname{diag}(\Delta_{21}(0,\beta)(s+1/3),\Delta_{21}(0,\beta),0)$. So
$\Delta_{21}=0$. But this can only happen if either
$\operatorname{ch}^{\beta}_{1}(v)=0=\operatorname{ch}^{\beta}_{1}(u)$ or we
are also on a $\nu$-wall. In the latter case we are in case (b), see below.
Otherwise $\Delta_{01}=0$. But then we either have the special situation
described before or $\Delta_{20}=0$ and $\Delta_{30}=0$ so we have $u\propto
v$. This is impossible. In the special case above we have $Hf_{u,v}=0$ again
and the same third derivatives as before are non-zero.

In case (b) above we have $\Delta_{21}=0$ and
$\operatorname{ch}^{\alpha,\beta}_{3}(u)=0=\operatorname{ch}^{\alpha,\beta}_{3}(u)$
so that $\Delta_{3i}=0$ for all $i$. Since it also crosses a $\nu$-wall we
must have $\Delta_{20}\neq 0$. When $\alpha\neq 0$, the local form in (2) can
then be read off the Hessian. Note that, at $s=1/3$ the wall has a standard
cusp singularity given by
$(\alpha-\alpha_{0})^{2}\propto(\beta-\beta_{0})^{3}$. Otherwise the Hessian
vanishes at the point. The third derivatives are then all zero except for
$\partial^{3}_{\beta}f_{u,v}=-2\Delta_{02}$. This gives the required form in
(3). ∎

######  Remark 6.8.

The special case where the surface is not regular for $\alpha>0$ splits into
two types: $v_{1}=0$ and $v_{1}\neq 0$. The former case gives $v_{3}=0=u_{3}$
and $u_{1}=0$. Then, so long as $\Delta_{02}\neq 0$, the surface has the local
form as stated in Theorem 6.7. It follows that
$f(\alpha,\beta,s)=((3s-1)\alpha^{2}+\beta^{2})\beta\Delta_{02}$, so
$\Sigma_{u,v}$ is reducible. As a concrete (actually generic) example,
consider the Chern character $v=(2,0,-3,0)$ and
$u=\operatorname{ch}({\mathcal{O}}_{X})$. On ${\mathbb{P}^{3}}$, $v$ is the
Chern character of rank 2 locally free sheaf $E$ with $c_{1}(E)=0$ and
$c_{2}(E)=3$, see Figure 6.

Figure 6. An example of a reducible surface $\Sigma_{u,v}$ with $u=(1,0,0,0)$
and $v=(2,0,-3,0)$, illustrating case (4) of Theorem 6.7.

On the other hand, if $v_{1}\neq 0$, the picture will be the same translated
by $\mu(v)$ along $\beta$. Note that (38) will determine $u_{1}$ for arbitrary
non-zero $\operatorname{ch}_{(}v)$ and $u_{0}$. Then (39) will determine
$v_{3}$, while (40) will determine $u_{2}$: one root is also a root of
$\Delta_{02}=0$ and so must be dismissed, and then $u_{2}$ is uniquely
determined. This gives a 3 parameter family of possible rational examples but
to correspond to actual objects there are strong diophantine constraints which
will depend on the threefold $X$.

######  Corollary 6.9.

If $\Delta_{01}\neq 0$ then $\Sigma_{u,v}\cap\\{\alpha>0,s>0,s\neq 1/3\\}$ is
regular and $\Upsilon_{u,v,s_{0}}=\Sigma_{u,v}\cap\\{s=s_{0}\\}$ is bounded
for each $s_{0}>0$.

A numerical $\lambda$-wall $\Upsilon_{\alpha,\beta,s}$ is said to be
_horizontal_ at a point $(\alpha_{0},\beta_{0})\in\mathbb{H}$ if
$\partial_{\beta}f_{u,v,s}(\alpha_{0},\beta_{0})=0$.

######  Proposition 6.10.

Assume $u$ and $v$ are as in Theorem 6.7. Let $s=1/3$. A numerical
$\lambda$-wall is horizontal at a point $(\alpha,\beta)\not\in\Theta_{v}$ if
and only if $(\alpha,\beta)\in\Gamma_{v,s}$ or $(\alpha,\beta)\in\Xi_{u,v}$
away from $\Theta_{v}$. It is a local maximum on $\Gamma_{v,s}$ and a local
minimum of $\Xi_{u,v}$ except for the special case where $\delta_{01}=0$ and
$(\alpha,\beta)\in\Gamma_{v,s}\cap\Xi_{u,v}$ where it is a point of
inflection.

###### Proof.

By Theorem 6.7, $\Upsilon_{\alpha,\beta,s}$ is horizontal if and only if
$\partial_{\beta}f_{u,v,s}=\Delta_{13}(\alpha,\beta)=0$. Assume first that the
point is on $\Gamma_{v,s}$ (but not on $\Theta_{v}$) so that
$\operatorname{ch}^{\alpha,\beta}_{3}(v)=0$. Then also
$\operatorname{ch}^{\alpha,\beta}_{3}(u)=0$ because
$\operatorname{ch}_{2}^{\alpha,\beta}(v)\neq 0$ and
$f_{u,v,s}(\alpha,\beta)=0$. Hence, $\Delta_{31}(\alpha,\beta)=0$ and so
$\partial_{\beta}f_{u,v,s}=0$ and the wall is horizontal. The second
derivative is then

$\frac{d^{2}\alpha}{d\beta^{2}}=-\frac{\partial^{2}_{\beta}f_{u,v,s}}{\partial_{\alpha}f_{u,v,s}}=\frac{\Delta_{21}+\Delta_{30}}{\alpha\Delta_{21}-\alpha\Delta_{30}}=-\frac{1}{\alpha}.$

If the point is on the associated $\nu$-wall $\Xi_{u,v}$, then, since
$\operatorname{ch}_{1}^{\beta}(v)\neq 0$, we have

$\Delta_{32}=\frac{\operatorname{ch}_{2}^{\alpha,\beta}(v)}{\operatorname{ch}_{1}^{\beta}(v)}\Delta_{31},$

and so if also $\Delta_{32}=0$ then the $\Upsilon_{u,v,s}$ must be horizontal
there, unless the point lies on $\Theta_{v}$. The second derivative (if the
point is not on $\Gamma_{v,s}$) is $1/\alpha$. Note that when the point is
also on $\Gamma_{v,s}$, the wall is singular by Theorem 6.7 item (2) and
otherwise $\Delta_{21}$ and $\Delta_{30}$ cannot vanish simultaneously except
in the special case where $\delta_{01}=0$. Consequently, there is never a
point of inflection on a numerical $\lambda$-wall except at
$\Gamma_{v,s}\cap\Xi_{u,v}$ in this special case.

Assume that $\Upsilon_{u,v,s}$ is horizontal at a point $(\alpha,\beta)$ not
on $\Gamma_{v,s}$. Then $\Delta_{13}(\alpha,\beta)=0$ and from
$\Delta_{23}(\alpha,\beta)=0$ we also have $\Delta_{12}(\alpha,\beta)=0$ by
Lemma 2.7(3) since $\operatorname{ch}_{3}^{\alpha,\beta}\neq 0$ by assumption.
So $(\alpha,\beta)$ is on the associated $\nu$-wall. ∎

######  Remark 6.11.

As a numerical $\lambda$-wall might be horizontal as it crosses $\Theta_{v}$,
though it generally is not. In that case, we can use $\beta$ and $s$ as local
coordinates on the surface and the second fundamental form at that point is
$\operatorname{II}=\frac{1}{\alpha}d\beta^{2}$. Then the Gauss curvature is
zero and the mean curvature is $1/2\alpha$.

Now we consider how numerical $\lambda$-walls vary with $s$. Note that
$\Sigma_{u,v}\cap\\{s=s_{0}\\}=\Upsilon_{u,v,s_{0}}$.

######  Theorem 6.12.

Let $\Sigma^{\prime}$ be a non-empty connected component of $\Sigma_{u,v}$.

  1. (1)

Suppose $s_{0}\geq 1/3$. If $\Sigma^{\prime}\cap\\{s=s_{0}\\}\neq\emptyset$,
then $\Sigma^{\prime}\cap\\{s=s_{1}\\}\neq\emptyset$ for every $s_{1}\geq
1/3$.

  2. (2)

Suppose $s_{0}<1/3$. If $\Sigma^{\prime}\cap\\{s=s_{0}\\}\neq\emptyset$, then
$\Sigma^{\prime}\cap\\{s=s_{1}\\}\neq\emptyset$ for every $0\leq s_{1}<s_{0}$.

If $\Sigma^{\prime}\cap\\{s=s_{0}\\}\neq\emptyset$, then for every
$0<s_{1}\leq s_{0}$, $\Sigma^{\prime}\cap\\{s=s_{1}\\}\neq\emptyset$.

In other words, if a numerical $\lambda$-wall exists for one $s_{0}\geq 1/3$,
then it must also exist for all $s\geq 1/3$, whereas if $s_{0}<1/3$, then it
need only exist for $s<s_{0}$. This means that numerical $\lambda$-walls can
only be “created” for $s<1/3$ and as $s$ decreases. We shall see below that
walls cannot be “created” in $R^{\pm}_{v,s}$ even when $s<1/3$ but they can be
created in $R^{0}_{v,s}$.

###### Proof.

Observe first that we may assume boundedness away from $s=0$, because if
$\Delta_{01}=0$, then $\Delta_{20}\neq 0$ and so $f_{u,v,s}(\alpha,\beta)=0$
is a cubic in $\beta$. Then it must have a solution for all $\alpha$. But if a
wall crosses $\alpha=0$ for one value of $s$, then it crosses for all $s$.

By Theorem 6.7, the surface is regular except at special points on $\alpha=0$
where there are multiple tangent planes which include the $s$ direction or we
are in case (2) of the theorem. But this latter case cannot arise in the
present situation. This means that if $\Sigma_{u,v}\cap\\{s=s_{1}\\}$ is empty
for some $s_{1}>0$, then there is some $s_{0}$ such that
$\partial_{\alpha}f_{u,v,s_{0}}=0=\partial_{\beta}f_{u,v,s_{0}}$ and
$\partial_{s}f_{u,v}|_{s=s_{0}}\neq 0$. It follows that $\alpha\neq 0$ and
$\Delta_{12}(\alpha,\beta)\neq 0$.

Then $\Sigma_{u,v}|_{s_{0}}$ is a union of closed curves, unbounded curves or
one or more distinct points. Suppose $(\alpha,\beta)$ is a point on a closed
curve component. Since $f$ is a quadratic function of $\alpha^{2}$ it follows
that nearby $(\alpha,\beta)$ there are two distinct solutions for a fixed
$\beta$. If the curve does not cross $\alpha=0$ then there are 4 distinct
solutions sufficiently close to two points in $\alpha>0$. But then there are
also 4 points for $\alpha<0$ which is impossible. So the curve must cross
$\alpha=0$. But this is a contradiction as $\alpha\neq 0$. We can also
eliminate the possibility of unbounded curves as follows. Each unbounded curve
must have two distinct unbounded branches. Then nearby $s_{0}$, there will be
four unbounded branches. But the implicit function defining the $\lambda$-wall
$\Upsilon_{u,v,s_{0}}$ is only asymptotically cubic (see the proof of
Proposition 4.4) and so this is impossible.

Consequently, $\Sigma_{u,v}\cap\\{s=s_{0}\\}$ consists only of isolated
points. Now consider one of these points, with coordinates
$(\alpha_{0},\beta_{0})$. From the regularity can use $\alpha$ and $\beta$ as
local coordinates on $\Sigma_{u,v}$ at this point. Note that the first
fundamental form at $(\alpha_{0},\beta_{0})$ is $d\alpha^{2}+d\beta^{2}$ from
the vanishing of $-\partial_{\alpha}f_{u,v}/\partial_{s}f_{u,v}$ and
$-\partial_{\beta}f_{u,v}/\partial_{s}f_{u,v}$.

From $f_{u,v}(\alpha_{0},\beta_{0},s_{0})=0$,
$\partial_{\alpha}f_{u,v}(\alpha_{0},\beta_{0},s_{0})=0$ and
$\partial_{\beta}f_{u,v}(\alpha_{0},\beta_{0},s_{0})=0$ we have

(42)
$\displaystyle\Delta_{32}=\alpha_{0}^{2}\left(s_{0}-1/3\right)\Delta_{12}$
(43)
$\displaystyle\Delta_{30}=\left(1+2\left(s_{0}-1/3\right)\right)\Delta_{21}+\alpha_{0}^{2}\left(s_{0}-1/3\right)\Delta_{10}$
(44)
$\displaystyle\Delta_{31}=\alpha_{0}^{2}\left(s_{0}-1/3\right)\Delta_{02},$

where we abbreviate $\Delta_{ij}(\alpha_{0},\beta_{0})$ to $\Delta_{ij}$ and
also for the rest of the proof. Together with Lemma 2.7 item (1), these
identities imply that

(45) $\alpha^{2}(s_{0}-1/3)\Delta_{02}^{2}=-(2s_{0}+1/3)\Delta_{12}^{2}.$

Since $\alpha\neq 0$ and $\Delta_{12}\neq 0$ we deduce that $\Delta_{02}\neq
0$ and $s<1/3$. This establishes the first part of the Theorem.

For the second part we show that the mean curvature at our point is negative
and so the surface must curve towards towards $s<s_{0}$. To this end observe
that the Gauss curvature $K$ is a positive multiple of
$\partial_{\alpha}^{2}f_{u,v}\partial_{\beta}^{2}f_{u,v}-(\partial_{\alpha}\partial_{\beta}f_{u,v})^{2}$.
Using (43) we have

$\displaystyle\partial_{\alpha}^{2}f_{u,v}$
$\displaystyle=2\alpha_{0}^{2}(2s_{0}+1/3)\Delta_{10}$
$\displaystyle\partial_{\alpha}\partial_{\beta}f_{u,v}$
$\displaystyle=2\alpha_{0}(s_{0}-1/3)\Delta_{02}$
$\displaystyle\partial_{\beta}^{2}f_{u,v}$
$\displaystyle=\alpha_{0}^{2}(s-1/3)\Delta_{10}+2(s_{0}+2/3)\Delta_{21}$

Using (45), we have
$(\partial_{\alpha}\partial_{\beta}f_{u,v})^{2}=-(2s_{0}+1/3)(s_{0}-1/3)\Delta_{21}^{2}$.
Then $K$ has the same sign as

$\begin{split}\alpha_{0}^{4}(s_{0}-1/3)\Delta_{10}^{2}+\alpha_{0}^{2}(s_{0}+2/3)\Delta_{10}\Delta_{21}+(s_{0}-1/3)\Delta_{21}^{2}=\\\
(s_{0}-1/3)(\alpha_{0}\Delta_{10}+\Delta_{12})^{2}+3s_{0}\alpha_{0}^{2}\Delta_{10}\Delta_{21}\end{split}$

But the tangent plane at $(\alpha_{0},\beta_{0})$ only intersects
$\Sigma_{u,v}$ at that point and so $K\geq 0$. The first term is negative and
so we must have $\Delta_{10}\Delta_{21}>0$.

The mean curvature is then

$-\frac{\partial_{\alpha}^{2}f_{u,v}}{\partial_{s}f_{u,v}}-\frac{\partial_{\beta}^{2}f_{u,v}}{\partial_{s}f_{u,v}}=-3s_{0}\alpha_{0}\frac{\Delta_{10}}{\Delta_{21}}-\frac{2}{\alpha_{0}}\left(s_{0}+\frac{2}{3}\right)<0$

as required. ∎

We observe that numerical $\lambda$-walls do admit isolated points, see Figure
7 below.

Figure 7. This is the surface wall $\Sigma_{u,v}$ for $v=(2,-1,-1,0)$ and
$u=(0,1,-10,-3)$. Taking $s_{0}\simeq 0.0569$, the point
$(\alpha,\beta)\simeq(6.24,3.52)$ is an isolated point of the numerical
$\lambda$-wall $\Upsilon_{u,v,s_{0}}=\Sigma_{u,v}|_{s_{0}}$.

As a consequence of Theorem 6.7 and the proof of Theorem 6.12 we can deduce
that 1-dimensional components of numerical $\lambda$-walls for a fixed $s$ are
regular away from $\nu$-walls.

######  Corollary 6.13.

For any $s>0$ and any real numerical Chern characters $u$ and $v$, any
connected component of a numerical $\lambda$-wall $\Upsilon_{u,v,s}$ in
$\mathbb{H}$ is regular as a real curve away from $\Xi_{u,v}$. In particular,
if there is no associated $\nu$-wall for the pair $u$, $v$, then
$\Upsilon_{u,v,s}$ is always regular.

###### Proof.

Away from the $\nu$-wall $\Xi_{u,v}$ (if non empty) when $s=1/3$,
$\Sigma_{u,v}$ is regular by Proposition 6.6 or Theorem 6.7. But by the proof
above, the only place where the plane tangent to $\Sigma_{u,v}$ is parallel to
the $(\alpha,\beta)$ plane occurs at isolated points and not in a one-
dimensional portion of the surface wall. ∎

We complete this section by returning to the issue of the intersection between
of numerical $\lambda$-walls for $v$ and the curves $\Theta_{v}$ and
$\Gamma_{v,s}$.

Unraveling the equality in (33) yields a cubic polynomial equation for
$\beta_{0}$. This means that the intersection
$\Upsilon_{u,v,s}\cap\Gamma_{v,s}$ consists of at most three points away from
$\Theta_{v}$; in addition, according to Lemma 4.11, the number of intersection
points does not depend on the parameter $s$. Notice that the total the number
of intersection points of a $\lambda$-wall for $v$ with $\Gamma_{v,s}$ will
increase by $1$ if $q(v)<0$.

Let us now examine one situation in which $\Upsilon_{u,v,s}\cap\Gamma_{v,s}$
contains two points.

######  Lemma 6.14.

If a connected component of a numerical $\lambda$-wall $\Upsilon_{u,v,s}$
crosses a connected component of $\Gamma_{v,s}$ twice away from the hyperbola
$\Theta_{v}$, then the associated numerical $\nu$-wall $\Xi_{u,v}$ is non-
empty.

###### Proof.

By the Lemma 4.11, we can let $s=1/3$. Then the two intersection points are
local maxima of $\Upsilon_{u,v,1/3}$. By Corollary 6.13, the component is
regular or has a tacnode on a $\nu$-wall. If it is regular then it must have a
minimum between the two maxima. Proposition 6.10 implies that such a minimum
must lie on the associated $\nu$-wall. ∎

Figure 8 illustrates the typical situation described in Lemma 6.14, and shows
that it does arise. It is easy to see that the intersection with $\alpha=0$
must happen to the right of $\Gamma^{-}_{v,s}$ because $\Gamma^{-}_{v,s}$ is
monotonic.

Figure 8. This example contains several interesting features. We set $s=1/3$,
$v=(3,1,0,-1)$ and $u=(0,1,-3,7)$. First, the curve $\Gamma_{v,s}$ (in red)
intersects the positive branch of the hyperbola $\Theta_{v}$ (in blue); this
intersection is marked with a black bullet . Second, the numerical
$\lambda$-wall $\Upsilon_{u,v,s}$ (in black) crosses the curve $\Gamma_{v,s}$
four times (marked with green bullets), twice along $\Gamma_{v,s}^{-}$.
Finally, we can see that the associated numerical $\nu$-wall $\Xi_{u,v}$ (in
magenta) cuts $\Gamma_{v,s}^{-}$ (red bullet) between the two intersection
points of $\Upsilon_{u,v,s}\cap\Gamma_{v,s}^{-}$, illustrating the phenomenon
described in Lemma 6.14. Both $\Upsilon_{u,v,s}$ and $\Xi_{u,v}$ cut
$\Theta_{v}^{-}$ at the same point.

The situation in Figure 8 also demonstrates another phenomenon in which a
numerical $\lambda$-wall intersects horizontal lines four times. When this
happens it must be that $f(\alpha,\beta)$ has three turning points along this
horizontal line. But note that $\partial_{\beta}f=-\Delta_{31}$ when $s=1/3$.
On the other hand, $\partial_{\alpha}\Delta_{31}=0$ and so the solutions of
$\Delta_{31}=0$ are vertical lines. But these intersect the wall at its
horizontal turning points (the green points in Figure 8). The middle one also
intersects $\Upsilon_{u,v,1/3}$ again at a minimum or on $\Theta_{v}$ which
must therefore also be on $\Xi_{u,v}$. For a connected component we already
could deduce this because such a component must have at least one minimum but
the same will follow even when the geometry of the wall does not require there
to be a minimum. This more precise reasoning allows us to refine Lemma 6.14:

######  Lemma 6.15.

If a numerical $\lambda$-wall $\Upsilon_{u,v,s}$ intersects $\Gamma^{-}_{v,s}$
twice, then $\Upsilon_{u,v,s}$ must intersect $\Theta_{v}$ and $\Xi_{u,v}$
intersects $\Gamma^{-}_{v,s}$ inbetween the intersection points.

###### Proof.

Note that Lemma 6.14 shows that if the wall is a connected component then it
must intersect $\Xi_{u.v}$. Since $\Delta_{31}=0$ are vertical lines there
must be minima above or below the maxima (the green dots on
$\Gamma^{-}_{v,1/3}$ in Figure 8). Since the wall cannot cross
$\Gamma^{-}_{v,s}$ for a third time because it would have to double back on
itself and there can only be at most two solutions of $\Delta_{32}=0$ along
any vertical line, it must cross $\Xi_{u,v}$ a third time at a point which is
not a minimum which must therefore be on $\Theta_{v}$. Note that an
alternative picture to Figure 8 has the wall cross $\alpha=0$ twice to the
left of $\Gamma^{-}_{v,s}$. In that case it only crosses $\Xi_{u,v}$ once and
then $\Xi_{u,v}$ does not intersect $\Gamma^{-}_{u,v}$ between the two
intersection points.

Now we assume the wall is two nested components. First we observe that this
hypothesis is independent of $s$. To see that note that if a component is
created at $(\overline{\alpha},\overline{\beta})$ then there would be $6$
distinct solutions of $f(\alpha,\beta)=0$ along $\beta=\overline{\beta}$ which
is impossible. Then these two bounded components must have maxima at $s=1/3$
which must intersect $\Gamma^{-}_{v,1/3}$.

Then there are three vertical line components of $\Delta_{31}=0$ which
intersect $\Gamma^{-}_{v,1/3}$ at the maxima of the wall at distinct $\beta$
values. Since the inner bounded component must have a maximum (on
$\Gamma^{-}_{v,1/3}$) the middle component of $\Delta_{31}=0$ intersects the
wall at that point and so intersects the outer component away from
$\Gamma^{-}_{v,1/3}$. But this point cannot be a point of inflection (see the
remark after Proposition 6.10) and so must be a minimum which must therefore
be on a $\nu$-wall. If the wall is otherwise in the exterior of the $\nu$-wall
then that point must be at the maximum of the $\nu$-wall and so is on
$\Theta_{v}$. Otherwise, it must cross the $\nu$-wall again which is not a
turning point and so must also cross $\Theta_{v}$. In fact this minimum of
$\Upsilon_{u,v,1/3}$ in the outer component must have another maximum which
must be on $\Gamma_{v}$ (see Figure 9 below for a concrete example of this
case). ∎

We can argue similarly to Lemma 6.15 in $R^{0}_{v}$:

######  Lemma 6.16.

If a numerical $\lambda$-wall $\Upsilon_{u,v,s}$ intersects $\Gamma^{0}_{v,s}$
twice, then $\Upsilon_{u,v,s}$ intersects $\Xi_{u,v}$ and $\Theta_{v}$.

###### Proof.

Observe first that by Lemma 4.11, the hypothesis and conclusion are
independent of $s$ and so we may set $s=1/3$.

If $\Upsilon_{u,v,s}$ has a single component then there must be a minimum
between the maxima and so it crosses $\Xi_{u,v}$ at that point. If there are
two components then the geometry does not require there to be a minimum. But
then there are three vertical components of $\Delta_{31}=0$. The middle one
intersects the inner component at its maximum and then it must intersect the
other component at a minimum (by Proposition 6.10) which must be on
$\Xi_{u,v}$.

In the last case, the outer component is either unbounded and the minimum
occurs on $\Theta_{v}$ or if it is bounded there is a further maximum which
must be on $\Gamma^{\pm}_{v.1/3}$ and so the wall again crosses $\Theta_{v}$.
The shape is illustrated in Figure 9. ∎

We can now state the main Theorem of this section:

######  Theorem 6.17.

Suppose a real numerical Chern character $v$ satisfies the Bogomolov
inequality and $v_{0}\neq 0$. Any connected bounded component of a numerical
$\lambda$-wall in $R^{-}_{v,s}$ for some $s\geq 1/3$ intersects
$\Gamma^{-}_{v,s}$.

###### Proof.

Let the component be $\Upsilon_{u,v,s}$ and suppose for a contradiction that
it does not cross $\Gamma^{-}_{v,s}$. By Theorem 6.12, this component must
exist for all $s$ and so we may assume $s=1/3$. By Theorem 6.7,
$\Upsilon_{u,v,1/3}$ is regular except possibly in case (2) of that Theorem,
in which case the singular point is on $\Gamma^{-}_{v,s}$ as required.
Otherwise $\Upsilon_{u,v,1/3}$ is regular and so it must have a maximum. By
Proposition 6.10, it must intersect $\Gamma^{-}_{v,1/3}$ at this point. ∎

######  Remark 6.18.

In other words, if we want to classify all of the actual $\lambda$-walls in
the region $R^{-}_{v,s}$ then we only need to locate the ones which cross
$\Gamma^{-}_{v,s}$. But note that the wall may not be actual as it crosses
$\Gamma^{-}_{v,s}$.

On the other hand, this fails for $s<1/3$ because isolated walls can appear as
$s$ decreases. Some will be unbounded as $s\to 0$ but others may persist as
isolated closed curves.

Finally, observe that analogous statements also hold for $\Gamma^{+}_{v,s}$
and the region to the right of it.

##  7\. Asymptotic $\lambda_{\alpha,\beta,s}$-stability in $R_{v,s}^{\pm}$

Similarly to Definition 5.1, we introduced the following definition, where
$\gamma:[0,\infty)\to\mathbb{H}$ be an unbounded path.

######  Definition 7.1.

An object $A\in D^{\rm b}(X)$ is _asymptotically
$\lambda_{\alpha,\beta,s}$-(semi)stable along $\gamma$_ if the following two
conditions hold for a given $s>0$:

  * (i)

there is $t_{0}>0$ such that $A\in\mathcal{A}^{\gamma(t)}$ for every
$t>t_{0}$;

  * (ii)

for every sub-object $F\hookrightarrow A$ in $\mathcal{A}^{\gamma(t)}$ with
$t\gg t_{0}$, there is $t_{1}>t_{0}$ such that
$\lambda_{\gamma(t),s}(F)<\leavevmode\nobreak\ (\leq)\leavevmode\nobreak\
\lambda_{\gamma(t),s}(A)$ for all $t>t_{1}$.

Our first goal is to characterize asymptotically
$\lambda_{\alpha,\beta,s}$-semistable objects with numerical Chern character
$v$ satisfying $v_{0}>0$ and the Bogomolov inequality along two families of
paths contained in the region $R_{v}^{-}$, namely the paths $\Gamma^{-}_{v,s}$
for each $s>0$, and $\Lambda^{-}_{\overline{\alpha}}$ for each
$\overline{\alpha}>0$, see the notation introduced in display (35).

We then invoke Proposition 5.13 to characterize asymptotically
$\lambda_{\alpha,\beta,s}$-semistable objects along the paths
$\Gamma^{+}_{v,s}$ and $\Lambda^{+}_{\overline{\alpha}}$. More precisely, we
show that $\lambda_{\alpha,\beta,s}$-semistable objects along
$\Gamma^{+}_{v,s}$ and $\Lambda^{+}_{\overline{\alpha}}$ are duals of Gieseker
semistable sheaves.

First, we provide a simple consequence of asymptotic
$\lambda_{\alpha,\beta,s}$-semistability and the support property along
unbounded paths.

######  Lemma 7.2.

Let $A\in D^{\rm b}(X)$ be an asymptotically
$\lambda_{\alpha,\beta,s}$-semistable object along and unbounded path
$\gamma(t)$. Then $Q^{\mathrm{tilt}}(A)\geq 0$.

###### Proof.

The support property in Proposition 2.4 implies that $Q_{\gamma(t)}(A)\geq 0$
for all large enough $t$.

Suppose first that $\lim_{t\to\infty}\beta(t)=\pm\infty$ then

$0\leq\lim_{t\to\infty}\frac{Q_{\gamma(t)}}{\beta(t)^{2}}=\left(\lim_{t\to\infty}\frac{\alpha(t)^{2}}{\beta(t)^{2}}+1\right)Q^{\mathrm{tilt}}(A).$

As the first factor is positive for all $t$, it follows that
$Q^{\mathrm{tilt}}(A)\geq 0$.

If $\lim_{t\to\infty}\alpha(t)=\infty$ then

$0\leq\lim_{t\to\infty}\frac{Q_{\gamma(t)}}{\alpha(t)^{2}}=Q^{\mathrm{tilt}}(A)$

and the same conclusion follows. ∎

###  7.1. Asymptotics along $\Gamma^{-}_{v,s}$

We now turn to asymptotically $\lambda_{\alpha,\beta,s}$-semistable objects
with Chern character $v$ (of an object) along the curve $\Gamma^{-}_{v,s}$; we
assume from now on that $v$ satisfies $v_{0}\neq 0$ and the Bogomolov
inequality (3).

Since $\operatorname{ch}_{1}^{\beta}(E)\neq 0$ along $\Gamma^{\pm}_{v,s}$, we
have:

(46) $\lambda_{\alpha,\beta,s}(E)=0\leavevmode\nobreak\ \leavevmode\nobreak\
\Leftrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\alpha^{2}=\dfrac{\operatorname{ch}^{\beta}_{3}(E)}{(s+1/6)\operatorname{ch}_{1}^{\beta}(E)},$

so we can use $\beta<0$ as a parameter.

Substituting for $\alpha^{2}$ as in equation (46) into the expression for
$\lambda_{\alpha,\beta,s}(u)$, we define the function
$\overline{\lambda}_{v,\beta,s}(u)$; to be precise

$\overline{\lambda}_{v,\beta,s}(u)=\dfrac{\operatorname{ch}_{3}^{\beta}(u)\operatorname{ch}_{1}^{\beta}(v)-\operatorname{ch}_{3}^{\beta}(v)\operatorname{ch}_{1}^{\beta}(u)}{\operatorname{ch}_{2}^{\beta}(u)\operatorname{ch}_{1}^{\beta}(v)-u_{0}\operatorname{ch}_{3}^{\beta}(v)/(2s+1/3)}$

expresses $\lambda$-slope of an object $F$ with $\operatorname{ch}(F)=u$ along
$\Gamma^{\pm}_{v,s}$. If $\operatorname{ch}_{0}(F)\neq 0$, the previous
expression yields

(47)
$\overline{\lambda}_{v,\beta,s}(F)=-\dfrac{6s+1}{9s\operatorname{ch}_{0}(E)\operatorname{ch}_{0}(F)}\cdot\dfrac{\delta_{01}\beta^{3}-3\delta_{02}\beta^{2}+3(\delta_{03}+\delta_{12})\beta-3\delta_{13}}{\beta^{3}+\textrm{
lower order terms}},$

where $\delta_{ij}=\delta_{ij}(F,E)$, as defined in equation (11). When
$\operatorname{ch}_{0}(F)=0$ and $\operatorname{ch}_{1}(F)\neq 0$, we obtain

(48)
$\overline{\lambda}_{v,\beta,s}(F)=-\dfrac{1}{3}\dfrac{\operatorname{ch}_{0}(E)\operatorname{ch}_{1}(F)\beta^{3}+\textrm{
lower order
terms}}{\operatorname{ch}_{0}(E)\operatorname{ch}_{1}(F)\beta^{2}+\textrm{
lower order terms}}$

Note that a numerical $\lambda$-wall $\Upsilon_{u,v,s}$ crosses
$\Gamma_{v,s}^{\pm}$ precisely at the zeros of the function
$\overline{\lambda}_{v,\beta,s}(u)$.

The behaviour of the $\nu$-slope of an object $F\in D^{\rm b}(X)$ along
$\Gamma_{v,s}^{\pm}$ can be analysed in a similar way. Substituting for
$\alpha^{2}$ as in equation (46) into the expression for
$\nu_{\alpha,\beta}(F)$, we obtain

(49)
$\lim_{\beta\to-\infty}\dfrac{-1}{\beta}\nu_{\alpha,\beta}(F)=\begin{cases}\dfrac{s}{2s+1/3},&\textrm{if
}\operatorname{ch}_{0}(F)\neq 0,\\\ 1,&\textrm{if
}\operatorname{ch}_{0}(F)=0,\leavevmode\nobreak\ \operatorname{ch}_{1}(F)\neq
0\end{cases}$

We are now in position to state the main result of this section.

######  Theorem 7.3.

Let $v$ be a numerical Chern character with $v_{0}\neq 0$ satisfying the
Bogomolov inequality (3). For each $s>0$, an object $A\in D^{\rm b}(X)$ with
$\operatorname{ch}(A)=v$ is asymptotically
$\lambda_{\alpha,\beta,s}$-(semi)stable along $\Gamma_{v,s}^{-}$ if and only
if $A$ is a Gieseker (semi)stable sheaf.

The proof will be done in a series of lemmas.

######  Lemma 7.4.

For every $s>0$, if there is $\beta_{0}<0$ such that
$E\in\mathcal{A}^{\alpha,\beta}$ for $(\alpha,\beta)\in\Gamma^{-}_{v,s}$ with
$\beta<\beta_{0}$, then $E\in\mathcal{C}oh(X)$.

This claim actually follows directly from Proposition 5.13 but we give an
alternative hands-on proof.

###### Proof.

Using the notation of Section 2.3, note that
$\nu_{\alpha,\beta}^{+}(E_{1})\leq 0$, since
$E_{1}\in\mathcal{F}_{\alpha,\beta}$, for every
$(\alpha,\beta)\in\Gamma^{-}_{v,s}$ with $\beta<\beta_{0}$. However, equation
(49) implies that for every sub-object $F\hookrightarrow E_{1}$ there exists
$\beta_{0}^{\prime}<0$, depending only on $\operatorname{ch}_{\leq 2}(F)$,
such that $\nu_{\alpha,\beta}(F)>0$ for every
$(\alpha,\beta)\in\Gamma^{-}_{v,s}$ and $\beta<\beta_{0}^{\prime}$, leading to
a contradiction. It follows that $E_{1}=0$.

On the other hand, the inequality

$\operatorname{ch}_{1}^{\beta}(E_{01})=\operatorname{ch}_{1}(E_{01})-\beta\operatorname{ch}_{0}(E_{01})\leq
0$

for all $\beta\ll 0$ implies that $\operatorname{ch}_{0}(E_{01})=0$, thus
$E_{01}=0$ since $E_{01}$ is torsion free. ∎

######  Lemma 7.5.

For every $s>0$, if $E$ is an asymptotically
$\lambda_{\alpha,\beta,s}$-(semi)stable object along $\Gamma^{-}_{v,s}$, then
$E$ is a Gieseker (semi)stable sheaf.

###### Proof.

If $E$ is asymptotically $\lambda_{\alpha,\beta,s}$-semistable along
$\Gamma^{-}_{v,s}$, then $E$ is a sheaf by Lemma 7.4 or Proposition 5.13.

If $E$ is not torsion free, let $F\hookrightarrow E$ be its maximal torsion
subsheaf; if $T\hookrightarrow F$ is a subsheaf of dimension $\leq 1$, then
$T\hookrightarrow E$ is a morphism in ${\mathcal{A}}^{\alpha,\beta}$ for
$(\alpha,\beta)\in\Gamma^{-}_{v,s}$ and $\beta\ll 0$, since
$T\in\mathcal{T}_{\alpha,\beta}$ for every $(\alpha,\beta)$. We have that
$\operatorname{ch}_{0}(T)=\operatorname{ch}_{1}(T)=0$, then

(50)
$\overline{\lambda}_{v,\beta,s}(T)=\begin{cases}-\beta+\dfrac{\operatorname{ch}_{3}(T)}{\operatorname{ch}_{2}(T)},&\textrm{if
}\operatorname{ch}_{2}(F)\neq 0,\\\ +\infty,&\textrm{if
}\operatorname{ch}_{2}(T)=0\end{cases}$

so $T$ would destabilize $E$. Therefore, we can assume that $F$ has pure
dimension 2; let $T$ be its maximal $\hat{\mu}$-semistable subsheaf. Remark
5.7 implies that $T\in{\mathcal{A}}^{\alpha,\beta}$ for
$(\alpha,\beta)\in\Gamma^{-}_{v,s}$ and $\beta\ll 0$, thus $T\hookrightarrow
E$ is a morphism in ${\mathcal{A}}^{\overline{\alpha},\beta}$ in the same
range. Since $\operatorname{ch}_{0}(T)=0$ and $\operatorname{ch}_{1}(T)\neq
0$, we have

$\lim_{\beta\to-\infty}\dfrac{-1}{\beta}\overline{\lambda}_{v,\beta,s}(F)=\dfrac{1}{3},$

again contradicting asymptotic $\lambda$-semistability. We therefore conclude
that $E$ must be torsion free.

If $F\hookrightarrow E$ is a proper (torsion free) subsheaf with
$\delta_{01}(F,E)\neq 0$, then equation (47) implies that

$\lim_{\beta\to-\infty}\overline{\lambda}_{v,\beta,s}(F)=-\dfrac{6s+1}{9s\operatorname{ch}_{0}(E)\operatorname{ch}_{0}(F)}\delta_{01}(F,E)\leq
0$

by hypothesis, thus $\delta_{01}(F,E)\geq 0$.

If $\delta_{01}(F,E)=0$, then equation (47) implies that

$\lim_{\beta\to-\infty}(-\beta)\overline{\lambda}_{v,\beta,s}(F)=-\dfrac{6s+1}{3s\operatorname{ch}_{0}(E)\operatorname{ch}_{0}(F)}\delta_{02}(F,E)\leq
0$

thus $\delta_{02}(F,E)\geq 0$.

Finally, if $\delta_{01}(F,E)=\delta_{02}(F,E)=0$, then

$\lim_{\beta\to-\infty}\beta^{2}\overline{\lambda}_{v,\beta,s}(F)=-\dfrac{6s+1}{3s\operatorname{ch}_{0}(E)\operatorname{ch}_{0}(F)}\delta_{03}(F,E)\leq
0$

thus $\delta_{03}(F,E)\geq 0$, with equality holding if and only of
$\delta_{03}(F,E)=0$ as well. In other words, $E$ is Gieseker (semi)stable. ∎

We now prove the converse of Lemma 7.5. The difficulty is that the strong
definition of asymptotic stability includes showing that a given object has
only finitely many walls along $\Gamma^{-}_{v,s}$. In fact we can show that
there is at most one, at least outside its actual $\nu$-wall.

Assume $E$ is a Gieseker stable sheaf and suppose there is an actual
$\lambda$-wall given by the short exact sequence $0\to F\to E\to G\to 0$
crossing $\Gamma^{-}_{v,s}$. Consider the case where $F$ and $G$ are sheaves.

######  Lemma 7.6.

Fix $s>0$, ${\overline{\beta}}$. Suppose $E$ is a Gieseker stable sheaf with
$\operatorname{ch}(E)=v$ and $F\hookrightarrow E$ is a subobject in both
$\mathcal{C}oh(X)$ and $\mathcal{A}^{\alpha,\beta}$ such that
$E/F\in\mathcal{A}^{\alpha,\beta}\cap\mathcal{C}oh(X)$ for all
$\beta<{\overline{\beta}}$ and $(\alpha,\beta)\in\Gamma^{-}_{v,s}$ then there
is some $\beta_{0}<{\overline{\beta}}$ such that
$\lambda_{\alpha,\beta,s}(F)<\lambda_{\alpha,\beta,s}(E)$ for all
$\beta<\beta_{0}$.

###### Proof.

We can compute the same limits as in the proof of Lemma 7.5; we first have

$\lim_{\beta\to-\infty}\overline{\lambda}_{\alpha,\beta,s}(F)=-\dfrac{6s+1}{9s\operatorname{ch}_{0}(E)\operatorname{ch}_{0}(F)}\delta_{01}(F,E)\leq
0,$

since $\delta_{01}(F,E)\geq 0$ by hypothesis. If $\delta_{01}(F,E)=0$, then

$\lim_{\beta\to-\infty}(-\beta)\overline{\lambda}_{\alpha,\beta,s}(F)=-\dfrac{6s+1}{3s\operatorname{ch}_{0}(E)\operatorname{ch}_{0}(F)}\delta_{02}(F,E)\leq
0$

because $\delta_{02}(F,E)\geq 0$. Finally, if $\delta_{02}(F,E)$ also
vanishes, then

$\lim_{\beta\to-\infty}\beta^{2}\overline{\lambda}_{\alpha,\beta,s}(F)=-\dfrac{6s+1}{3s\operatorname{ch}_{0}(E)\operatorname{ch}_{0}(F)}\delta_{03}(F,E)\leq
0$

since $\delta_{03}(F,E)\geq 0$, with equality holding if and only if
$\delta_{03}(F,E)=0$ as well, which implies that
$\overline{\lambda}_{\alpha,\beta,s}(F)=0$ for every
$(\alpha,\beta)\in\Gamma^{-}_{v,s}$ for $\beta\ll 0$. ∎

This enables us to prove

######  Lemma 7.7.

Suppose $E$ is a Gieseker stable sheaf and $F\hookrightarrow E$ is a subsheaf
which also corresponds to an actual $\lambda$-wall $W_{u,v,s}$ crossing
$\Gamma^{-}_{v,s}$ at a point $P$ beyond any $\nu$-wall for $E$. If
$E/G\in\mathcal{A}^{\alpha,\beta}\cap\mathcal{C}oh(X)$ then $E$ must be stable
above the point $P$.

###### Proof.

Parametrize the curve $\Gamma^{-}_{v,s}$ by $\gamma(t)$ outwards with
$\gamma(0)=P$. Suppose $E$ is unstable in $\mathcal{A}^{\gamma(t)}$ for
$\epsilon>t>0$. Observe that for $t\gg 0$ we have that $F\hookrightarrow E$ is
a monomorphism. This is because If
$\lambda_{\gamma(t),s}(F)\geq\lambda_{\gamma(t),s}(E)$ for all $t>0$ then
Corollary 5.14 and Lemma 7.6 gives a contradiction. It follows that at some
point $Q$ above $P$, we have $\lambda_{Q,s}(F)=\lambda_{Q,s}(E)$ and so the
numerical $\lambda$-wall $\Upsilon_{u,v,s}$ crosses $\Gamma^{-}_{v,s}$ twice.
Then Lemma 6.15 tells us that there is a numerical $\nu$-wall $\Xi_{u,v}$
intersecting $\Gamma^{-}_{v,s}$ between $P$ and $Q$. But $0\to F\to E\to G\to
0$ is a short exact sequence in $\mathcal{B}^{\beta(t)}$ for all $t\geq 0$ and
so this is an actual $\nu$-wall contradicting the assumption. ∎

We now show that above an actual $\nu$-wall for $E$, the $\lambda$-wall
equivalent of Lemma 5.3(2) holds:

######  Theorem 7.8.

Suppose $E$ is a Gieseker stable sheaf with Chern character $v$ then there is
at most one actual wall $W_{u,v,s}$ intersecting $\Gamma^{-}_{v,s}$ at a point
$P$ above the actual $\nu$-wall for $E$. Moreover such a wall destabilizes
downwards along $\Gamma^{-}_{v,s}$.

###### Proof.

Consider first a wall $W_{u,v,s}$, given by a short exact sequence $0\to A\to
E\to B\to 0$, which intersects $\Gamma^{-}_{v,s}$ at a point $P$. Note that
beyond $P$ along $\Gamma^{-}_{v,s}$, $E$ is $\nu$-stable. Denote the path from
$P$ along $\Gamma^{-}_{v,s}$ by $\gamma$ with $\gamma(0)=P$. Note that $A$ and
$B$ are $\lambda_{P,s}$-semistable. If both $A$ and $B$ are sheaves then Lemma
7.7 shows that the wall destabilizes downwards. More generally,
$A\in\mathcal{B}^{\gamma(t)}$ for all $t\geq 0$. If $A$ is not a sheaf then by
Proposition 2.12, $E=A_{00}$ and $B=A_{01}[2]$. If $A_{01}[1]$ is not
$\nu$-semistable. Let $Q$ be the Harder–Narasimhan factor of least
$\nu$-slope. Then $\nu^{+}(Q)<0$ so $Q[1]\in\mathcal{A}^{P}$ and by the
octahedral axiom we have a $\mathcal{B}^{\beta(0)}$ short exact sequence $Q\to
D\to A_{00}$ where $Q$ is a $\mathcal{B}^{\gamma(0)}$ quotient of $A$. Then
$D\to E\to Q[1]$ is also destabilizing. So we may assume $A_{01}[1]$ is
$\nu$-stable. But then, along $\gamma$, $A_{01}[2]$ remains in
$\mathcal{A}^{\gamma(t)}$ until we hit the
$\Theta^{+}_{\operatorname{ch}(A_{01})}$ (we must hit it because $A_{01}$
satisfies the Bogomolov inequality). But at that point
$\lambda(A_{01})=+\infty$. So again the wall must cross $\Gamma^{-}_{v,s}$
again and so there is a numerical $\nu$-wall. But this is impossible as
$\nu(A_{01})<0<\nu(A_{00})$ to the right of
$\Theta^{+}_{\operatorname{ch}(A_{01})}$.

So we may assume $A\in\mathcal{C}oh(X)$. As in the proof of Proposition 2.12,
we can split the exact $\mathcal{B}^{\beta(0)}$ sequence

$0\to B_{1}\to A\to E\to B_{0}\to 0$

via
$Q\in\mathcal{A^{\gamma(0)}}\cap\mathcal{B}^{\beta(0)}\cap\mathcal{C}oh(X)$.
Proceeding as before with $B_{1}$, let $B^{+}$ be the factor with the highest
$\nu_{P}$. Then $G:=B/B^{+}[1]\in\mathcal{A}^{P}$ and we have a
$\mathcal{A}^{P}$ short exact sequences $N\to E\to G$ and $A\to N\to B^{+}[1]$
in $\mathcal{A}^{P}$. Since $\lambda(B^{+})\to\infty$ as we approach
$\Theta^{-}_{\operatorname{ch}(B^{+})}$ along $\gamma$, we see that
$\lambda(N)>\lambda(E)=0$ near $\Theta^{-}_{\operatorname{ch}(B^{+})}$ while
$\lambda_{P,s}(N)\leq 0$. So we may replace $A$ with $N$ (which is again a
sheaf by Proposition 2.12). Repeating we may assume $B_{1}$ is $\nu$-stable.
Then as we approach $\Theta^{-}_{\operatorname{ch}(B_{1})}$,
$\lambda(B_{1})\to\infty$ so we could replace $B$ with $B_{0}$. So we may
assume $B\in\mathcal{B}^{\beta(0)}$. If we assume first that $B_{00}\neq 0$
then we can apply the same argument above to assume that $B_{01}$ is
$\nu$-stable and then since $\lambda(B_{01})\to+\infty$ as we approach
$\Theta^{-}_{\operatorname{ch}(B_{01})}$, it follows that $F:=A/B_{01}$ in
$\mathcal{C}oh(X)$ destabilizes $E$ with quotient $B_{00}$ (this is case (1)
in Proposition 2.12). So we may assume $A$ and $B$ are sheaves and we have
shown that $E$ is $\lambda$-stable above $P$.

So if we have two walls $W_{u,v,s}$ and $W_{u^{\prime},v,s}$ corresponding to
$F_{1}\to E$ and $F_{2}\to E$ crossing $\Gamma^{-}_{v,s}$ at $P=\gamma(0)$ and
$Q=\gamma(1)$, say, respectively. Then $F_{2}$ destabilizes $E$ below $Q$. If
$F_{2}\to E$ remains an injection in $\mathcal{A}^{\gamma(t)}$ for $0\leq
t\leq 1$, $W_{u^{\prime},v,s}$ must cross $\Gamma^{-}_{v,s}$ again and then
there would be an actual $\nu$-wall between $P$ and $Q$ which is not permitted
by hypothesis. So it must be that $F_{2}\to E$ ceases to be an injection at
some point. Let that point be $R$. We show that $E$ remains unstable as we
cross $R$. In fact we will prove a stronger result in the following lemma. ∎

######  Lemma 7.9.

Suppose $E$ is a Gieseker stable sheaf with $\operatorname{ch}(E)=v$ and let
$\gamma$ be a curve segment of $\Gamma^{-}_{v,s}$ from a point $Q=\gamma(0)$
to $P=\gamma(1)$ which is outside an actual $\nu$-wall. Suppose $0\to A\to
E\to B\to 0$ is a short exact sequence in $\mathcal{A}^{P}$ corresponding to
an actual $lambda$-wall through the point $Q$. Then $E$ is
$\lambda_{\gamma(t),s}$-unstable for all $t\in(0,1]$.

###### Proof.

By the second statement of the previous lemma we know that $0\to A\to E\to
B\to 0$ destabilizes for $t\in(0,\epsilon)$ for some $\epsilon>0$. We also
know from that proof that $\lambda_{\gamma(t),s}(A)>\lambda_{\gamma(t),s}(E)$
for all $t\in(0,1]$. In the case where $A$ is not a sheaf we have that $0\to
A\to E\to B\to 0$ remains short exact to the end of $\Gamma^{-}_{v,s}$ as
$E=A_{00}$ and $B=A_{01}[2]$ and $\Gamma^{-}_{v,s}$ ends either on the
$\beta$-axis or $\Theta^{-}_{v}$. So we may assume $A$ is a sheaf.

Then $A$ remains in the category until it’s Harder–Narasimhan factor with
smallest $\nu$, $A^{-}$, say, goes out of the category. But then just beyond
that point $\lambda(A^{-})>0$ and so $E\to\ker_{\mathcal{A}}(B\to A^{-}[1])$
still destabilizes $E$. We can continue until we have exhausted all of the
Harder–Narasimhan factors of $A$. Finally, as we approach the last
$\Theta^{-}_{\operatorname{ch}(A^{\prime})}$ curve for the filtration of $A$
we have $\lambda(A^{\prime})<0$. But then there would be a wall corresponding
to $A^{\prime}$ destabilizing $E$ above which is impossible. It follows that
$A$ remains in $\mathcal{A}^{\gamma(t)}$ for all $t\in[0,1]$.

Similarly, $B_{0}$ remains in $\mathcal{A}^{\gamma(t)}$ until $\Theta^{-}$ of
a factor but this can’t happen as $E$ remains in the category. But then $B$
remains in the category by Theorem 3.3. This completes the proof. ∎

We can now deduce the converse to Lemma 7.5.

######  Lemma 7.10.

If $E$ is a Gieseker (semi)stable sheaf then $E$ is asymptotically
$\lambda_{\alpha,\beta,s}$-(semi)stable along $\Gamma^{-}_{v,s}$.

###### Proof.

If $E$ is Gieseker stable then the first part of Definition 7.1 follows from
the fact that $E\in\mathcal{C}oh(X)$ and Proposition 5.13 and the second now
from Theorem 7.8. The statement for semistability follows by inducting on the
length of the Jordan–Hölder filtration of $E$. ∎

######  Remark 7.11.

Just as for Proposition 5.8, we can deduce that any
$E\in\mathcal{A}^{\alpha,\beta}$ for all $(\alpha,\beta)$ along an unbounded
$\Theta^{-}$ curve $\gamma(t)$ in $R^{-}_{v}$ has an asymptotic
Harder–Harasimhan filtration for $\lambda_{\alpha\,\beta,s}$-stability from
Lemma 7.10.

The following will be useful in the next section.

######  Proposition 7.12.

If $E$ is a Gieseker stable sheaf with $\operatorname{ch}(E)=v$ and
$W_{u,v,s}$ is an actual $\lambda$-wall in $R^{-}_{v}$ crossing
$\Gamma^{-}_{v,s}$. Suppose the underlying numerical $\lambda$-wall crosses
$\Theta^{-}_{v}$ at a point $P$. Then there is an actual $\nu$-wall for $E$ at
or above this point of intersection.

###### Proof.

By Lemma 4.17, $W_{u,v,s}$ ends either on another actual $\lambda$-wall which
must remain above the original wall. We can repeat with the new wall. We claim
that this terminates after a finite number of steps with an actual
$\lambda$-wall crossing $\Theta^{-}_{v}$ above $P$. Assume otherwise. Then as
the piecewise curve $T$ of walls is trapped between $\Gamma^{-}_{v,s}$ and
$\Theta^{-}_{v}$, the curve must be unbounded. Therefore there is some
$\overline{\alpha}$ such that $({\overline{\beta}},\overline{\alpha})\in T$
for some ${\overline{\beta}}$ and the line
$({\overline{\beta}}+t,\overline{\alpha})$ is above the largest $\lambda$-wall
crossing $\Theta^{-}_{v}$. Then by Proposition 4.18,
$({\overline{\beta}}+t,\overline{\alpha})$ must cross an actual $\lambda$-wall
corresponding to the same object. So the underlying numerical $\lambda$-wall
loops back. By changing $s$ either the wall vanishes (when $s<1/3$) and then
the process must terminate and the wall crosses $\Theta^{-}_{v}$ for all $s$
as required, or at $s=1/3$ it crosses $\Gamma^{-}_{v,1/3}$ at its turning
point which is impossible. ∎

###  7.2. Asymptotics along $\Lambda_{\overline{\alpha}}^{-}$

Next, we study objects that are asymptotically
$\lambda_{\alpha,\beta,s}$-semistable along a horizontal line
$\\{\alpha=\overline{\alpha}\\}$, or _asymptotically
$\lambda_{\overline{\alpha},\beta,s}$-semistable_. The key is to observe that
$\Lambda^{-}_{\overline{\alpha}}$ is eventually in $R^{L}_{v,s}$ and we can
use Theorem 6.17. We establish the following result.

######  Theorem 7.13.

Let $v$ be a Chern character of an object of $D^{\rm b}(X)$ satisfying
$v_{0}\neq 0$ and the Bogomolov inequality (3). For each $s\geq 1/3$ and each
$\overline{\alpha}>0$, an object $A\in D^{\rm b}(X)$ is asymptotically
$\lambda_{\alpha,\beta,s}$-semistable objects along
$\Lambda_{\overline{\alpha}}^{-}$ if and only if $A$ is a Gieseker semistable
sheaf.

The strategy of the proof is very similar to the one used in Theorem 7.3,
though with different calculations; we include them here for the sake of
completeness. Indeed, note that

(51)
$\lim_{\beta\to-\infty}\frac{-1}{\beta}\nu_{\overline{\alpha},\beta}(F)=\begin{cases}1/2,&\text{if
}\operatorname{ch}_{0}(F)\neq 0;\\\ 1,&\text{if
}\operatorname{ch}_{0}(F)=0\text{ and }\operatorname{ch}_{1}(F)\neq
0.\end{cases}$

As before, the claim follows from item (1) of Proposition 5.13; alternatively,
it can also be proved in the same manner as Lemma 7.4 above.

######  Lemma 7.14.

Fix $\overline{\alpha}>0$. If there is $\beta_{0}<0$ such that
$E\in\mathcal{A}^{\overline{\alpha},\beta}$ for every $\beta<\beta_{0}$, then
$E\in\mathcal{C}oh(X)$.

The next step is to understand the difference in $\lambda$-slopes; note that

(52)
$\lambda_{\alpha,\beta,s}(F)-\lambda_{\alpha,\beta,s}(E)=\dfrac{f_{u,v}(\alpha,\beta)}{\rho_{u}(\alpha,\beta)\rho_{v}(\alpha,\beta)}$

with $u=\operatorname{ch}(F)$ and $v=\operatorname{ch}(E)$. The full
expression of numerator is given by equation (30); the numerator is given by

$\rho_{u}(\alpha,\beta)\rho_{v}(\alpha,\beta)=\dfrac{1}{4}\operatorname{ch}_{0}(F)\operatorname{ch}_{0}(E)\beta^{4}-\dfrac{1}{2}(\operatorname{ch}_{1}(F)\operatorname{ch}_{0}(E)+\operatorname{ch}_{1}(E)\operatorname{ch}_{0}(F))\beta^{3}$
$+\left(\dfrac{1}{2}\operatorname{ch}_{0}(F)\operatorname{ch}_{0}(E)\alpha^{2}+\operatorname{ch}_{1}(F)\operatorname{ch}_{1}(E)+\dfrac{1}{2}(\operatorname{ch}_{0}(F)\operatorname{ch}_{2}(E)+\operatorname{ch}_{2}(F)\operatorname{ch}_{0}(E))\right)\beta^{2}$
$+(\textrm{ terms of lower order in }\beta).$

With these formulas at hand, we can establish the _if_ part of Theorem 7.13.
First we prove a version of Lemma 7.6 along $\Lambda^{-}_{\overline{\alpha}}$.

######  Lemma 7.15.

For any fixed $s>0$ and $\overline{\alpha}>0$, if $E$ is a Gieseker stable
sheaf then there is no monomorphism $F\hookrightarrow E$ in
$\mathcal{A}^{\overline{\alpha},\beta}$ with
$\lambda_{\overline{\alpha},\beta,s}(F)\geq\lambda_{\overline{\alpha},\beta,s}(E)$
for all $\beta<{\overline{\beta}}$.

###### Proof.

If $F\hookrightarrow E\twoheadrightarrow G$ is a short exact sequence in
$\mathcal{A}^{\overline{\alpha},\beta}$ for every $\beta<\beta_{0}$, then
Lemma 7.14 implies that both $F$ and $G$ are sheaves. We first have that

$\lim_{\beta\to-\infty}\left(\lambda_{\overline{\alpha},\beta,s}(F)-\lambda_{\overline{\alpha},\beta,s}(E)\right)=\dfrac{1}{3}\dfrac{\delta_{10}(F,E)}{\operatorname{ch}_{0}(F)\operatorname{ch}_{0}(E)}\leq
0,$

since $\delta_{10}(F,E)\leq 0$ by hypothesis. If $\delta_{10}(F,E)=0$, then
also

$\lim_{\beta\to-\infty}(-\beta)(\lambda_{\overline{\alpha},\beta,s}(F)-\lambda_{\overline{\alpha},\beta,s}(E))=\dfrac{4}{3}\dfrac{\delta_{20}(F,E)}{\operatorname{ch}_{0}(F)\operatorname{ch}_{0}(E)}\leq
0,$

because $\delta_{20}(F,E)\leq 0$. Finally, if $\delta_{20}(F,E)$ also
vanishes, then

$\lim_{\beta\to-\infty}\beta^{2}(\lambda_{\overline{\alpha},\beta,s}(F)-\lambda_{\overline{\alpha},\beta,s}(E))=4\dfrac{\delta_{30}(F,E)}{\operatorname{ch}_{0}(F)\operatorname{ch}_{0}(E)}\leq
0,$

since $\delta_{30}(F,E)\leq 0$, with equality holding if and only if
$\delta_{30}(F,E)=0$ as well, which implies that
$\lambda_{\overline{\alpha},\beta,s}(F)=\lambda_{\overline{\alpha},\beta,s}(E)$
for every $\overline{\alpha}$, $\beta$ and $s$. It follows that $E$ is
asymptotically $\lambda_{\overline{\alpha},\beta,s}$-(semi)stable, as desired.
∎

Now we extend Theorem 7.8 to $\Lambda^{-}_{\overline{\alpha}}$. The trick is
to reduce to the case of $\Gamma^{-}_{v,s}$.

######  Lemma 7.16.

Suppose $s\geq 1/3$ and suppose $E$ is a Gieseker semistable sheaf with
$\operatorname{ch}(E)=v$. Then there is a ${\overline{\beta}}$ such that for
all $\beta<{\overline{\beta}}$, there are no actual $\lambda$-walls
intersecting $\Lambda^{-}_{\overline{\alpha}}$. Furthermore, there are no more
actual $\lambda$-walls to the left of the actual $\nu$-wall for $E$ in
$R^{-}_{v}$.

###### Proof.

Suppose $W_{u,v,s}$ is an actual $\lambda$-wall intersecting
$\Lambda^{-}_{\overline{\alpha}}$ at a point
$P=(\beta_{0},\overline{\alpha})$. By choice of ${\overline{\beta}}$ we can
assume $P\in R^{L}_{v,s}$. By Theorem 6.12 any numerical $\lambda$-wall
$W_{u,v,s}$ for $s\geq 1/3$ remains a numerical $\lambda$-wall at $s=1/3$ and,
since it is bounded, each component must intersect $\Gamma^{-}_{v,1/3}$ at its
maxima. Arguing as in Theorem 7.8, this wall must destabilize to the right. If
$W_{u,v,1/3}$ is not actual as it crosses $\Gamma^{-}_{v,1/3}$ then by Lemma
4.17, it ends on another actual $\lambda$-wall. Repeating we have a piecewise
path of walls $W^{(i)}_{u,v,1/3}$ which must all destabilize to the right and
so the path has increasing $\alpha$. Then it must cross $\Gamma^{-}_{v,1/3}$.
But the actual $\lambda$-wall component $W^{N}_{v,1/3}$ crossing
$\Gamma^{-}_{v,1/3}$ must be the unique wall by Theorem 7.8. So there cannot
be another such wall to the left of $P$. Hence, we can let
${\overline{\beta}}=\beta_{0}$.

For the last sentence observe that if the wall crosses
$\Lambda^{-}_{\overline{\alpha}}$ again then there must be a piecewise smooth
curve of actual $\lambda$-walls which intersects $\Gamma^{-}_{v,s}$ at a point
$Q$, say. This can only happen if there is a $\nu$-wall above $Q$. ∎

This completes the proof of:

######  Lemma 7.17.

For any $s\geq 1/3$ and $\overline{\alpha}>0$, if $E$ is a Gieseker
(semi)stable sheaf, then $E$ is asymptotically
$\lambda_{\overline{\alpha},\beta,s}$-(semi)stable along
$\Lambda^{-}_{\overline{\alpha}}$.

We now consider the converse.

######  Lemma 7.18.

For any $s>0$ and $\overline{\alpha}>0$, if $E$ is asymptotically
$\lambda_{\overline{\alpha},\beta,s}$-(semi)stable along
$\Lambda^{-}_{\overline{\alpha}}$, then $E$ is a Gieseker (semi)stable sheaf.

###### Proof.

If $E$ is asymptotically $\lambda_{\alpha,\beta,s}$-(semi)stable along
$\alpha=\overline{\alpha}$, then Lemma 7.14 implies that $E$ is a sheaf.

If $E$ is not torsion free, let $F\hookrightarrow E$ be its maximal torsion
subsheaf; if $T\hookrightarrow F$ is a subsheaf of dimension 1, then
$T\hookrightarrow E$ is a morphism in
${\mathcal{A}}^{\overline{\alpha},\beta}$ for $\beta\ll 0$, since
$T\in\mathcal{T}_{\alpha,\beta}$ for every $(\alpha,\beta)$. We have that
$\operatorname{ch}_{0}(T)=\operatorname{ch}_{1}(T)=0$, thus

$\lim_{\beta\to-\infty}\dfrac{-1}{\beta}\left(\lambda_{\overline{\alpha},\beta,s}(T)-\lambda_{\overline{\alpha},\beta,s}(E)\right)=\dfrac{2}{3}>0,$

provided $\operatorname{ch}_{2}(T)\neq 0$, contradicting asymptotic
$\lambda$-semistability. If $\operatorname{ch}(T)=(0,0,0,e)$, then it clearly
destabilizes $E$ as well.

So now assume that $F$ has pure dimension 2, and let $T$ be its maximal
$\hat{\mu}$-semistable subsheaf. Remark 5.7 implies that
$T\in{\mathcal{A}}^{\overline{\alpha},\beta}$ for $\beta\ll 0$, thus
$T\hookrightarrow E$ is a morphism in
${\mathcal{A}}^{\overline{\alpha},\beta}$ in the same range. Since
$\operatorname{ch}_{0}(T)=0$ and $\operatorname{ch}_{1}(T)\neq 0$, then

$\lim_{\beta\to-\infty}\dfrac{-1}{\beta}\left(\lambda_{\overline{\alpha},\beta,s}(T)-\lambda_{\overline{\alpha},\beta,s}(E)\right)=\dfrac{1}{6}>0,$

again contradicting asymptotic $\lambda$-semistability. We therefore conclude
that $E$ is torsion free.

If $E$ is not Gieseker semistable, let $F\hookrightarrow E$ be its maximal
destabilizing subsheaf. As in the first part of the proof of Lemma 7.17, we
can conclude that $F\in{\mathcal{A}}^{\overline{\alpha},\beta}$ for $\beta\ll
0$, so $F\hookrightarrow E$ is a morphism in
${\mathcal{A}}^{\overline{\alpha},\beta}$ in the same range.

It follows that

$\lim_{\beta\to-\infty}\left(\lambda_{\overline{\alpha},\beta,s}(F)-\lambda_{\overline{\alpha},\beta,s}(E)\right)=\dfrac{1}{3}\dfrac{\delta_{10}(F,E)}{\operatorname{ch}_{0}(F)\operatorname{ch}_{0}(E)}\geq
0,$

thus $\delta_{10}(F,E)=0$ because $E$ is asymptotically
$\lambda_{\overline{\alpha},\beta,s}$-semistable. We then have that

$\lim_{\beta\to-\infty}(-\beta)(\lambda_{\overline{\alpha},\beta,s}(F)-\lambda_{\overline{\alpha},\beta,s}(E))=\dfrac{4}{3}\dfrac{\delta_{20}(F,E)}{\operatorname{ch}_{0}(F)\operatorname{ch}_{0}(E)}\geq
0,$

thus again $\delta_{20}(F,E)=0$. Finally, we have that

$\lim_{\beta\to-\infty}\beta^{2}(\lambda_{\overline{\alpha},\beta,s}(F)-\lambda_{\overline{\alpha},\beta,s}(E))=4\dfrac{\delta_{30}(F,E)}{\operatorname{ch}_{0}(F)\operatorname{ch}_{0}(E)}\geq
0,$

thus $\delta_{30}(F,E)=0$, meaning that
$\operatorname{ch}(F)=\operatorname{ch}(E)$, contradicting the fact that $F$
is a proper subsheaf of $E$. We therefore conclude that $E$ must be Gieseker
(semi)stable, as desired. ∎

###  7.3. Asymptotics along general unbounded $\Theta^{-}$-curves

We can now extend our results to any unbounded $\Theta^{-}$-curve (recall
Definition 5.9).

######  Theorem 7.19.

Let $\gamma$ be an unbounded $\Theta^{-}$-curve and fix $s\geq 1/3$. An object
$E\in D^{b}(X)$ is asymptotically $\lambda_{\alpha,\beta,s}$-(semi)stable
along $\gamma$ if and only if $E$ is Gieseker (semi)stable.

###### Proof.

We consider the stable case first and we deduce the semistable case by
inducting on the lengths of the Jordan–Hölder filtrations.

Suppose $E$ is Gieseker stable. Then Lemma 7.16 as $\overline{\alpha}$ varies
implies that there is at most a single destabilizing piecewise smooth curve of
actual $\lambda$-walls from the $\beta$ axis. It cannot be unbounded as for
large enough $\alpha$ using the same argument as in the proof of Proposition
7.12. So the piecewise curve must either cross the $\beta$ axis again or
$\Theta^{-}_{v}$. In either case there is is a ${\overline{\beta}}$ such that
$E$ is $\lambda_{\beta,\alpha,s}$-stable in
$\\{\beta<{\overline{\beta}}\\}\cap R^{-}_{v}$ and so asymptotically
$\lambda$-stable alongf any curve in this region.

Conversely, if $E$ is asymptotically $\lambda$-stable along $\gamma$ and
suppose it is not Gieseker stable. Then for all $t\gg 0$, $E$ is not
$\lambda$-stable along $\Lambda^{-}_{\alpha(t)}$. Let $t_{0}$ be the least
$t\in\mathbb{R}$ such that $E$ is $\lambda$-semistable in
$\mathcal{A}^{\gamma(t)}$ for all $t<t_{0}$. By increasing $t_{0}$ if
necessary, we may assume that for each $t>t_{0}$ there is some $F_{t}$ such
that $F_{t}$ destabilizes $E$ at some $\beta=\beta_{1}<\beta(t)$ along
$\Lambda^{-}_{\alpha(t)}$ and such that $\nu^{-}_{\gamma(t)}(F_{t})>0$ (by the
assumption that $\gamma$ is a $\Theta^{-}$-curve and Lemma 5.12). So there
must be a wall $W_{\operatorname{ch}(E),\operatorname{ch}(F_{t}),s}$ which
crosses the vertical line $\beta=\beta_{1}$. But that wall must cross either
$\Lambda^{-}_{\alpha(t)}$ at $\beta<\beta_{1}$ or $Im(\gamma)$ at
$\gamma(t^{\prime})$ for some $t^{\prime}>t$. Since it remains an actual
$\lambda$-wall, the latter possibility cannot happen by the assumption on
$t_{0}$. The former possibility means that there must be another
$F^{\prime}_{t}$ which destabilizes at some $(\alpha(t),\beta_{2})$ with
$\beta_{2}<\beta_{1}$. Repeating we have an infinite sequence of such
destabilizers which contradicts Lemma 7.16. ∎

######  Remark 7.20.

We expect that the assumption $s\geq 1/3$ is unnecessary. But there are
instances of walls which vanish as $s$ increases. These could occur to the
left of our piecewise destabilizing wall. For any $s$ we would conjecture that
there are only finitely many of these and then there would be a Gieseker
chamber containing a finite number of small regions where $E$ is
$\lambda$-unstable.

We can now deduce asymptotic conditions along unbounded curves in $R^{+}_{v}$.
Using the notion of unbounded $\Theta^{-}$-curve $\gamma$ and of its dual
curve $\gamma^{*}$ proposed in Definition 5.9, we state:

######  Proposition 7.21.

Let $\gamma$ be an unbounded $\Theta^{-}$-curve. Then $E$ is asymptotically
$\lambda_{\alpha,\beta,s}$-(semis)table along $\gamma$ if and only if
$E^{\vee}$ is asymptotically $\lambda_{\alpha,\beta,s}$-(semi)stable along
$\gamma^{*}$.

###### Proof.

By Proposition 5.13, we know that there is a $t_{1}$ such that
$E^{\vee}\in\mathcal{A}^{\gamma^{*}(t)}$ for all $t>t_{1}$. If
$F\hookrightarrow E$ destabilizes $E$ for all $t>t_{0}$ then the proposition
also shows that there is some $t_{2}$ such that $F^{\vee}$ and $E^{\vee}$ are
in $\mathcal{A}^{\gamma^{*}(t)}$ for all $t>t_{2}$ and also $E^{\vee}\to
F^{\vee}$ surjects in $\mathcal{A}^{\gamma^{*}(t)}$. But
$\lambda_{\gamma(t),s}(F)=-\lambda_{\gamma^{*}(t),s}(F^{\vee})$. ∎

Putting Theorem 7.19 and Proposition 7.21 together, we immediately deduce the
following statement.

######  Theorem 7.22.

Let $v$ be a numerical Chern character satisfying $v_{0}\neq 0$ and $Q^{\rm
tilt}(v)\geq 0$, and fix $\overline{\alpha}>0$. For each $s\geq 1/3$, an
object $A\in D^{\rm b}(X)$ with $\operatorname{ch}(A)=v$ is asymptotically
$\lambda_{\alpha,\beta,s}$-(semi)stable along an unbounded $\Theta^{+}$-curve
if and only if $A^{\vee}$ is a Gieseker (semi)stable sheaf.

Since Theorem 7.3 holds for every $s>0$, we have a stronger statement for the
asymptotics along $\Gamma^{+}_{v,s}$.

######  Theorem 7.23.

Let $v$ be a numerical Chern character satisfying $v_{0}\neq 0$ and $Q^{\rm
tilt}(v)\geq 0$. For each $s>0$, an object $A\in D^{\rm b}(X)$ is
asymptotically $\lambda_{\alpha,\beta,s}$-(semi)stable objects along
$\Gamma^{+}_{v,s}$ if and only if $A^{\vee}$ is a Gieseker (semi)stable sheaf.

Note that Proposition 2.15 provides an explicit characterization of objects
$A\in D^{\rm b}(X)$ that are dual to torsion free sheaves.

##  8\. Examples of classifying walls: ideal sheaves and null correlation
sheaves

We complete this paper by studying some concrete examples of actual
$\lambda$-walls and asymptotically $\lambda_{\alpha,\beta,s}$-stable objects
for $X={\mathbb{P}^{3}}$ which illustrate many of the results established
above.

###  8.1. Ideal sheaves of lines in ${\mathbb{P}^{3}}$

Let $L\subset{\mathbb{P}^{3}}$ be a line, and $I_{L}$ denote its ideal sheaf;
recall that $v:=\operatorname{ch}(I_{L})=(1,0,-1,1)$. The curves $\Theta_{v}$
and $\Gamma_{v,s}$ are given by

$\displaystyle\Theta_{v}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ :\leavevmode\nobreak\ \leavevmode\nobreak\
\beta^{2}-\alpha^{2}=2$ $\displaystyle\Gamma_{v,s}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ :\leavevmode\nobreak\
\leavevmode\nobreak\
\left(s+\dfrac{1}{6}\right)\beta\alpha^{2}-\dfrac{\beta^{3}}{6}+\beta=-1$

Note that $\Gamma_{v,s}$ intersects $\Theta_{v}^{-}$ for every $s>0$ at a
single point, call it $P_{s}$, whose $\beta$ coordinate is the real root of
the polynomial $3s\beta^{3}+(2-6s)\beta+3=0$; the case $s=1/3$ is pictured in
Figure 2.

Theorem 4.22 guarantees the existence of vanishing $\nu$\- and
$\lambda$-walls, which are precisely given by the triangle

${\mathcal{O}}_{\mathbb{P}^{3}}(-1)^{\oplus 2}\to
I_{L}\to{\mathcal{O}}_{\mathbb{P}^{3}}(-2)[1].$

We simplify the notation and use $\Xi:=\Xi_{u,v}$ and
$\Upsilon_{s}:=\Upsilon_{u,v,s}$, where
$u=\operatorname{ch}({\mathcal{O}}_{\mathbb{P}^{3}}(-1))$, for the vanishing
$\nu$\- and $\lambda$-walls just described, respectively.

Both curves cut $\Theta_{v}^{-}$ at the point
$R:=\Theta_{v}\cap\Xi\cap\Upsilon_{s}=(\alpha=1/2,\beta-3/2)$. In addition,
the vanishing $\lambda$-wall crosses $\Gamma_{v,s}^{-}$ at the point
$Q_{s}:=\Gamma_{v,s}^{-}\cap\Upsilon_{s}=(1/\sqrt{6s+1},-2)$.

There are no other actual $\nu$-walls for the numerical Chern vector
$v=(1,0,-1,1)$. To see that, suppose $A\to I_{L}$ is a
$\nu_{\alpha,\beta}$-stable destabilizing object in $\mathcal{B}^{\beta}$ then
it must destabilize along $\beta=-\sqrt{2}$ and also on $\Theta_{v}^{-}$. Let
$\operatorname{ch}_{\leq 2}(A)=(r,-c,d)$ then $r>0$ and the condition that
both $A$ and $I_{L}/A\in\mathcal{B}^{\beta}$ gives us

$\displaystyle-c-\beta r$ $\displaystyle<0$ $\displaystyle c-\beta(1-r)$
$\displaystyle\leq 0$

and so $-\beta r<c\leq\beta(1-r)$. For $\beta=-\sqrt{2}$ we see that $c=r$ is
the only solution. Then the first inequality implies
$r<\beta/(1+\beta)=\sqrt{2}/(\sqrt{2}-1)$ and so $0<r\leq 3$. Now choosing
$\beta$ on $\Theta_{v}^{-}$, we have $d+\beta c+r=0$ in order for $A$ to
destabilize $I_{L}$. So $d=-r(\beta+1)>r(\sqrt{2}-1)$. On the other hand, the
Bogomolov inequality for $A$ gives $r^{2}\geq 2rd$ and so $d\leq r/2$. Then
$d=r/2$ as it must be in $\mathbb{Z}[1/2]$.

Figure 9. The vanishing $\nu$\- and $\lambda$-walls (in purple and black,
respectively) for the ideal sheaf of a line. Note that the $\lambda$-wall is
connected but possesses two irreducible components. Both curves intersect
$\Theta_{v}$ (in blue) an $\Gamma_{v,s}^{-}$ (in red).

Since, ${\mathcal{O}}_{\mathbb{P}^{3}}(-2)$ and
${\mathcal{O}}_{\mathbb{P}^{3}}(-1)$ are $\lambda_{\alpha,\beta,s}$-stable for
all $(\alpha,\beta,s)$, it follows that his is the only actual $\lambda$-wall
destabilizing ideal sheaves of lines.

###  8.2. Ideal sheaves of points in ${\mathbb{P}^{3}}$

Let $p\in{\mathbb{P}^{3}}$ be a point, and $I_{p}$ denote its ideal sheaf;
recall that $v:=\operatorname{ch}(I_{p})=(1,0,0,-1)$. The curves $\Theta_{v}$
and $\Gamma_{v,s}$ are given by

$\displaystyle\Theta_{v}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ :\leavevmode\nobreak\ \leavevmode\nobreak\
\beta^{2}-\alpha^{2}=0$ $\displaystyle\Gamma_{v,s}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ :\leavevmode\nobreak\
\leavevmode\nobreak\
\left(s+\dfrac{1}{6}\right)\beta\alpha^{2}-\frac{\beta^{3}}{6}=1$

Note that $\Gamma_{v,s}$ does not intersect $\Theta_{v}^{-}$. There are no
actual $\nu$-walls because $\operatorname{ch}_{\leq
2}(I_{p})=\operatorname{ch}_{\leq 2}({\mathcal{O}}_{\mathbb{P}^{3}})$ and
$I_{p}\to{\mathcal{O}}_{\mathbb{P}^{3}}$ injects in
$\mathcal{A}^{\alpha,\beta}\cap\mathcal{B}^{\beta}\cap\mathcal{C}oh({\mathbb{P}^{3}})$.

The resolution of $I_{p}$

$0\to{\mathcal{O}}_{\mathbb{P}^{3}}(-3)\to{\mathcal{O}}_{\mathbb{P}^{3}}(-2)^{\oplus
3}\to{\mathcal{O}}_{\mathbb{P}^{3}}(-1)^{\oplus 3}\to I_{p}\to 0$

induces two triangles: the first one defines a reflexive sheaf $S_{p}$ with
$u:=\operatorname{ch}(S_{p}[1])=(-2,3,-3/2,-1/2)$ whose only singularity is
precisely the point $p$ by splitting the resolution in the middle. Then we
have triangles:

(53) ${\mathcal{O}}_{\mathbb{P}^{3}}(-2)^{\oplus 3}[1]\to
S_{p}[1]\to{\mathcal{O}}_{\mathbb{P}^{3}}(-3)[2];$

the second one relates the sheaf $S_{p}$ with $I_{p}$

(54) ${\mathcal{O}}_{\mathbb{P}^{3}}(-1)^{\oplus 3}\to I_{p}\to S_{p}[1].$

This provides one wall which cuts $\Gamma_{v,s}$ at $(1/\sqrt{6s+1},-2)$.
There is another wall which crosses at the same point given by an object $D$
which is not a sheaf (or a shift of a sheaf) with
$u^{\prime}:=\operatorname{ch}(D)=(0,3,-9/2,7/2)$.

$D\to I_{p}\to{\mathcal{O}}_{\mathbb{P}^{3}}(-3)[2]$

where $D$ is the cone on the morphism
${\mathcal{O}}_{\mathbb{P}^{3}}(-2)^{\oplus
3}\to{\mathcal{O}}_{\mathbb{P}^{3}}(-1)^{\oplus 3}$ in the resolution of
$I_{p}$. This gives rise to a fourth triangle:

${\mathcal{O}}_{\mathbb{P}^{3}}(-1)^{\oplus 3}\to
D\to{\mathcal{O}}_{\mathbb{P}^{3}}(-2)^{\oplus 3}[1].$

Note that this provides as example in Proposition 2.12 where
$D\not\in\mathcal{C}oh(X)$ and note that
$E/D\in\mathcal{C}oh({\mathcal{O}}_{\mathbb{P}^{3}})[2]\cap\mathcal{B}^{\beta}[1]$.
The objects ${\mathcal{O}}_{\mathbb{P}^{3}}(-3)[2]$,
${\mathcal{O}}_{\mathbb{P}^{3}}(-2)[1]$ and
${\mathcal{O}}_{\mathbb{P}^{3}}(-1)$ all belong to
$\mathcal{A}^{\alpha,\beta}$ for every $(\alpha,\beta)$ within a square with
vertices at the points $(\alpha=0,\beta=-2)$, $(\alpha=1/2,\beta=-3/2)$,
$(\alpha=1,\beta=-2)$, $(\alpha=1/2,\beta=-5/2)$. Consequently this also
follows for $S_{p}$ and $D$.

Note that at the special point $P$,
$\lambda_{\alpha,\beta,s}({\mathcal{O}}_{\mathbb{P}^{3}}(-1))=\lambda_{\alpha,\beta,s}({\mathcal{O}}_{\mathbb{P}^{3}}(-2))=\lambda_{\alpha,\beta,s}({\mathcal{O}}_{\mathbb{P}^{3}}(-3))=\lambda_{\alpha,\beta,s}(D)=\lambda_{\alpha,\beta,s}(S_{p})=\lambda_{\alpha,\beta,s}(I_{p})$.

Within this region, the two triangles induce actual $\lambda$-walls
$\Upsilon_{w,u,s}$, where
$w:=\operatorname{ch}({\mathcal{O}}_{\mathbb{P}^{3}}(-2)[1])$, and
$\Upsilon_{u,v,s}$. One can check that $\Upsilon_{w,u,s}$ and
$\Upsilon_{u,v,s}$ intersect at the point $(\alpha=1/\sqrt{6s+1},\beta=-2)$,
which also belongs to $\Gamma_{v,s}^{-}$, for every $s$, implying that the
sheaf $I_{p}$ is a $\lambda_{\alpha,\beta,s}$-semistable object at that point;
see Figure 10 below. It is easy to see that
$\operatorname{Ext}^{1}(D,{\mathcal{O}}_{\mathbb{P}^{3}}(-3)[2])=0=\operatorname{Ext}^{1}({\mathcal{O}}_{\mathbb{P}^{3}}(-1)^{\oplus
3},S_{p})$ and so the walls are vanishing.

Figure 10. This graph shows the numerical $\lambda$-walls $\Upsilon_{u,w,s}$
(in blue), $\Upsilon_{w^{\prime},u^{\prime},s}$ (in purple where
$w^{\prime}=\operatorname{ch}({\mathcal{O}}_{\mathbb{P}^{3}}(-1))$),
$\Upsilon_{w,v,s}$ (in black) and $\Upsilon_{u^{\prime},v,s}$ (in orange), and
the curve $\Gamma_{v,s}^{-}$ (in red) crossing at the point
$(\alpha=1/\sqrt{3},\beta=-2)$; we set $s=1/3$. The shaded region marks where
the objects $O(-3)[2]$, $O(-2)[1]$ and $O(-1)$, and consequently also
$S_{p}[1]$, $D$ and $I_{p}$, belong to $\mathcal{A}^{\alpha,\beta}$.

It follows that there exists an open set $V\subset\mathbb{H}$ containing the
point $(\alpha=1/\sqrt{6s+1},\beta=-2)$ in its boundary such that $I_{p}$ is a
$\lambda_{\alpha,\beta,s}$-stable object for every $(\alpha,\beta)\in V$.
Moreover, $\Upsilon_{u,v,s}$ is a vanishing $\lambda$-wall for $I_{p}$ for
$\beta>-2$ and $\Upsilon_{u^{\prime}v,s}$ is the vanishing wall for
$\beta<-2$. The resulting wall is $C^{1}$ on $\Gamma_{v,1/3}^{-}$ but only
$C^{0}$ for other values of $s$.

Since $I_{p}$ is also asymptotically $\lambda_{\alpha,\beta,s}$-stable along
$\Gamma_{v,s}^{-}$, it follow that $I_{p}$ is
$\lambda_{\alpha,\beta,s}$-stable for every
$(\alpha,\beta)\in\Gamma_{v,s}^{-}$ with $\beta<-2$ (that is, no actual
$\lambda$-wall crosses $\Gamma_{v,s}^{-}$ for $\beta<-2$).

###  8.3. Torsion free sheaves with Chern character $\mathbf{(2,0,-1,0)}$

Our first step towards the classification of
$\lambda_{\alpha,\beta,s}$-semistable objects with Chern character
$v=(2,0,-1,0)$ on ${\mathbb{P}^{3}}$ near the curve $\Gamma_{v,s}^{-}$ is the
classification of $\mu$-semistable torsion free sheaves with this Chern
character.

Below, $p$ and $L$ respectively denote a point and a line in
${\mathbb{P}^{3}}$. Recall also that torsion free $E$ on ${\mathbb{P}^{3}}$ is
called a _null correlation sheaf_ if it satisfies the exact sequence

$0\to{\mathcal{O}}_{\mathbb{P}^{3}}(-1)\to\Omega^{1}_{{\mathbb{P}^{3}}}(1)\to
E\to 0,$

see [12]. Note that $\operatorname{ch}(E)=(2,0,-1,0)$. A locally free null
correlation sheaf is called a null correlation bundle; non locally free null
correlation sheaves satisfy a sequence of the form

$0\to E\to{\mathcal{O}}_{\mathbb{P}^{3}}^{\oplus 2}\to\mathcal{O}_{L}(1)\to 0$

for some line $L\subset{\mathbb{P}^{3}}$.

######  Proposition 8.1.

Let $E$ be a $\mu$-semistable torsion free sheaf on $X={\mathbb{P}^{3}}$ with
$\operatorname{ch}(E)=(2,0,-1,0)$.

  1. (1)

If $E$ is locally free, then $E$ is a null correlation bundle; in particular,
$E$ is $\mu$-stable.

  2. (2)

If $E$ is properly torsion free, then $E$ is strictly $\mu$-semistable and it
is given by one of the following extensions:

     * (2.1)

$0\to I_{L}\to E\to I_{p}\to 0$ for $p\in L$ with nontrivial extension; in
particular, $E$ is a null correlation sheaf and it is $\mu_{\leq 2}$-stable;

     * (2.2)

$0\to I_{p}\to E\to I_{L}\to 0$ for arbitrary $p$ and $L$; in particular, $E$
is not $\mu_{\leq 2}$-semistable and it has no global sections;

     * (2.3)

$0\to{\mathcal{O}}_{\mathbb{P}^{3}}\to E\to I_{\tilde{L}}\to 0$, where
$\tilde{L}$ is a 1-dimensional scheme satisfying the sequence
$0\to\mathcal{O}_{p}\to\mathcal{O}_{\tilde{L}}\to\mathcal{O}_{L}\to 0$ for
arbitrary $p$ and $L$; in particular, $E$ is not $\mu_{\leq 2}$-semistable and
it has a global section.

Note that ${\rm Ext}^{1}(I_{L},I_{p})=0$ when $p\notin L$.

###### Proof.

If $E$ is locally free, then [11, Lemma 2.1] implies that $E$ is $\mu$-stable;
the fact that every $\mu$-stable rank 2 bundle $E$ with $c_{2}(E)=1$ on
${\mathbb{P}^{3}}$ is a null correlation bundle is proved in [29, 4.3.2, page
363].

If $E$ is not locally free, then $E^{**}$ is a $\mu$-semistable rank 2
reflexive sheaf with $\operatorname{ch}_{1}(E^{**})=0$ and either
$\operatorname{ch}_{2}(E^{**})=0$ or $\operatorname{ch}_{2}(E^{**})=1$; in
both cases, $E^{**}$ is strictly $\mu$-semistable (cf. [11, Lemma 2.1]), so
$E$ is strictly $\mu$-semistable. The first case forces
$E^{**}={\mathcal{O}}_{\mathbb{P}^{3}}^{\oplus 2}$, while, in the second case,
$E^{**}$ must be a properly reflexive sheaf $S_{L}$ given by the sequence

(55) $0\to{\mathcal{O}}_{\mathbb{P}^{3}}\to S_{L}\to I_{L}\to 0;$

note that $\operatorname{ch}(S_{L})=(2,0,-1,1)$.

We start by analysing the first case, that is
$E^{**}={\mathcal{O}}_{\mathbb{P}^{3}}^{\oplus 2}$. We have that
$\operatorname{ch}(Q_{E})=(0,0,1,0)$ where $Q_{E}:=E^{**}/E$; note that
$h^{0}(E)=0,1$. Again, two possibilities follow: either $Q_{E}$ has pure
dimension 1, in which case $Q_{E}=\mathcal{O}_{L}(1)$, or $Q_{E}$ satisfies a
sequence of the form

$0\to Z\to Q_{E}\to O_{L}(k)\to 0,$

where $Z$ is a 0-dimensional sheaf, and $k-1=-h^{0}(Z)<0$ because
$\operatorname{ch}_{3}(O_{L}(k))=-\operatorname{ch}_{3}(Z)$; since
$h^{0}(O_{L}(k))>0$, we must have $k=0$ and thus $Z=\mathcal{O}_{p}$.

The first possibility, namely $Q_{E}=\mathcal{O}_{L}(1)$, leads to the
sequence

$0\to E\to{\mathcal{O}}_{\mathbb{P}^{3}}^{\oplus 2}\to\mathcal{O}_{L}(1)\to 0$

and therefore yields the sheaves described in item (2.1), since $I_{L}$ is the
kernel of the composition
${\mathcal{O}}_{\mathbb{P}^{3}}\to{\mathcal{O}}_{\mathbb{P}^{3}}^{\oplus
2}\twoheadrightarrow\mathcal{O}_{L}(1)$. In addition, notice that these are
the null correlation sheaves. Checking that these sheaves are $\mu_{\geq
2}$-stable is a simple exercise.

The second possibility leads to the sequence

$0\to E\to{\mathcal{O}}_{\mathbb{P}^{3}}^{\oplus
2}\to\mathcal{O}_{\tilde{L}}\to 0$

and therefore yields the sheaves described in item (2.3).

Finally, if $E^{**}=S_{L}$, then $Q_{E}=\mathcal{O}_{p}$, and the sequence
$0\to E\to S_{L}\to\mathcal{O}_{p}\to 0$ together with the sequence (55) lead
to the sequence in item (2.2). ∎

It is important to notice that nontrivial extensions like the ones in items
(2.2) and (2.3) of Proposition 8.1 do exist. Let us first consider the
extension of an ideal sheaf of a line by an ideal sheaf of a point.

######  Lemma 8.2.

If $p$ is a point and $L$ is a line in ${\mathbb{P}^{3}}$, then

$\dim\operatorname{Ext}^{1}(I_{L},I_{p})=\left\\{\begin{array}[]{l}3\leavevmode\nobreak\
{\rm if}\leavevmode\nobreak\ p\notin L;\\\ 5\leavevmode\nobreak\ {\rm
if}\leavevmode\nobreak\ p\in L.\end{array}\right.$

###### Proof.

First, apply the functor $\operatorname{Hom}(-,I_{p})$ to the exact sequence
$0\to I_{L}\to{\mathcal{O}}_{\mathbb{P}^{3}}\to\mathcal{O}_{L}\to 0$ to
conclude that
$\operatorname{Ext}^{1}(I_{L},I_{p})\simeq\operatorname{Ext}^{2}(\mathcal{O}_{L},I_{p})$.
Applying the functor $\operatorname{Hom}(\mathcal{O}_{L},-)$ to the exact
sequence $0\to I_{p}\to{\mathcal{O}}_{\mathbb{P}^{3}}\to\mathcal{O}_{p}\to 0$
we obtain

(56)
$0\to\operatorname{Ext}^{1}(\mathcal{O}_{L},\mathcal{O}_{p})\to\operatorname{Ext}^{2}(\mathcal{O}_{L},I_{p})\to
H^{1}(\mathcal{O}_{L}(-4))\to\operatorname{Ext}^{1}(\mathcal{O}_{L},\mathcal{O}_{p})\to
0.$

Note that
$\operatorname{Ext}^{i}(\mathcal{O}_{L},\mathcal{O}_{p})=H^{0}({\mathcal{E}}{\it
xt}^{i}(\mathcal{O}_{L},\mathcal{O}_{p}))$. If $p\notin L$, then
${\mathcal{E}}{\it xt}^{i}(\mathcal{O}_{L},\mathcal{O}_{p})=0$ for every
$i\geq 0$, thus $\operatorname{Ext}^{1}(I_{L},I_{p})\simeq
H^{1}(\mathcal{O}_{L}(-4))$, which completes the proof of the first claim.

If $p\in L$, then one can check that

$h^{0}({\mathcal{E}}{\it
xt}^{i}(\mathcal{O}_{L},\mathcal{O}_{p}))=\left\\{\begin{array}[]{l}2\leavevmode\nobreak\
{\rm for}\leavevmode\nobreak\ i=1;\\\ 1\leavevmode\nobreak\ {\rm
for}\leavevmode\nobreak\ i=2.\end{array}\right.$

Comparing with the exact sequencein display (56), we obtain the second part of
the statement. ∎

Next we consider the sheaves in item (2.3) of Proposition 8.1.

######  Lemma 8.3.

If $\tilde{L}$ is the 1-dimensional scheme described in item (2.3) of
Proposition 8.1, then
$\dim\operatorname{Ext}^{1}(I_{\tilde{L}},{\mathcal{O}}_{\mathbb{P}^{3}})=1$.

###### Proof.

Note that

$\operatorname{Ext}^{1}(I_{\tilde{L}},{\mathcal{O}}_{\mathbb{P}^{3}})\simeq
H^{2}(I_{\tilde{L}}(-4))^{*}\simeq H^{2}(I_{L}(-4))^{*},$

where the first isomorphism is given by Serre duality, and the second follows
from the sequence $0\to I_{\tilde{L}}\to I_{L}\to\mathcal{O}_{p}\to 0$.

The resolution of the ideal sheaf $I_{L}$ yields the cohomology sequence

$0\to H^{2}(I_{L}(-4))\to H^{3}({\mathcal{O}}_{\mathbb{P}^{3}}(-6))\to
H^{3}({\mathcal{O}}_{\mathbb{P}^{3}}(-5))^{\oplus 2}\to H^{3}(I_{L}(-4))\to
0;$

however,
$H^{3}(I_{L}(-4))\simeq\operatorname{Hom}(I_{L},{\mathcal{O}}_{\mathbb{P}^{3}})^{*}$,
thus $h^{2}(I_{L}(-4))=1$. ∎

Now let $\mathcal{G}$ denote the Grassmanian of lines in ${\mathbb{P}^{3}}$.
Recall that a line $L\in\mathcal{G}$ is called a _jumping line_ for a
$\mu$-semistable torsion free sheaf $E$ with $\operatorname{ch}_{1}(E)=0$ on
${\mathbb{P}^{3}}$ if
$E\otimes\mathcal{O}_{L}=\mathcal{O}_{L}(-a)\oplus\mathcal{O}_{L}(a)$ for some
$a>0$; we denote by $\mathcal{J}(E)$ the set of jumping lines for $E$, as a
subset of $\mathcal{G}$.

If $E$ is a null correlation sheaf, then $\mathcal{J}(E)$ is a divisor of
degree 1 in $\mathcal{G}$. Therefore, for each null correlation sheaf $E$ and
$L\in\mathcal{J}(E)$ there exists an epimorphism
$E\twoheadrightarrow\mathcal{O}_{L}(-1)$, whose kernel is a Gieseker
semistable sheaf $K$ with $\operatorname{ch}(K)=(2,0,-2,2)$.

Such sheaves have been previously considered by Miró-Roig and Trautmann in
[28, Section 1.5]; they showed that the family of sheaves $K$ defined by exact
sequences of the form

$0\to K\to E\to\mathcal{O}_{L}(-1)\to 0$

where $E$ is a null correlation sheaf and $L\in\mathcal{J}(E)$, define a
locally closed 8-dimensional subscheme of the full moduli space of Gieseker
semistable sheaves on ${\mathbb{P}^{3}}$ with Chern character equal to
$(2,0,-2,2)$. In addition, this moduli space, which we will simply denote by
$\mathcal{Z}$, is an irreducible projective variety of dimension 13, and every
such sheaf satisfies an exact sequence of the form

(57) $0\to{\mathcal{O}}_{\mathbb{P}^{3}}(-2)^{\oplus 2}\stackrel{{\scriptstyle
A}}{{\to}}{\mathcal{O}}_{\mathbb{P}^{3}}^{\oplus 4}(-1)\to K\to 0,$

with $A$ being a $4\times 2$ matrix of linear forms in 4 variables.

Below it will be important to describe the set of pairs
$(K,L)\in\mathcal{Z}\times\mathcal{G}$ for which non-trivial extensions of $K$
by $\mathcal{O}_{L}(-1)$ exist; we consider the set

(58) $\mathcal{J}:=\\{(K,L)\in\mathcal{Z}\times\mathcal{G}\leavevmode\nobreak\
|\leavevmode\nobreak\ \operatorname{Ext}^{1}(K,\mathcal{O}_{L}(-1))\neq 0\\}.$

######  Lemma 8.4.

$\mathcal{J}$ is a divisor in $\mathcal{Z}\times\mathcal{G}$. In addition, the
natural projection $\mathcal{J}\to\mathcal{Z}$ is surjective, and its fibres
are quadric hypersurfaces in $\mathcal{G}$.

In particular, $\mathcal{J}$ is an irreducible projetive variety of dimension
16.

###### Proof.

Applying the functor $\operatorname{Hom}(-,\mathcal{O}_{L}(-1))$ to sequence
in display (57) we obtain

$0\to\operatorname{Hom}(K,\mathcal{O}_{L}(-1))\to
H^{0}(\mathcal{O}_{L})^{\oplus 4}\to H^{0}(\mathcal{O}_{L}(1))^{\oplus
2}\to\operatorname{Ext}^{1}(K,\mathcal{O}_{L}(-1))\to 0,$

thus
$\dim\operatorname{Ext}^{1}(K,\mathcal{O}_{L}(-1))=\dim\operatorname{Hom}(K,\mathcal{O}_{L}(-1))$
for every $L\in\mathcal{G}$.

However, $\operatorname{Hom}(K,\mathcal{O}_{L}(-1))\neq 0$ precisely when $L$
is a jumping line for $K$. According to [28, Lemma 3.10.1], for each
$K\in\mathcal{Z}$ the set of jumping lines for $K$ is a divisor of degree 2 in
$\mathcal{G}$. ∎

Finally, observe also that
$\dim\operatorname{Ext}^{1}(K,\mathcal{O}_{L}(-1))\leq 4$, depending on the
rank of the map $H^{0}(\mathcal{O}_{L})^{\oplus 4}\to
H^{0}(\mathcal{O}_{L}(1))^{\oplus 2}$ which is given by the restriction of the
matrix $A$ to the line $L$.

Our next goal in this section is the classification of
$\nu_{\alpha,\beta}$-stable objects for the Chern character $v=(-2,0,1,0)$.
First, we show that there are no $\nu$-walls.

######  Lemma 8.5.

For $v=\pm(2,0,-1,k)$ there are no actual $\nu$-walls for any $k$.

###### Proof.

We suppose first that $\beta<0$. Then we may assume $E$ is a Gieseker
semistable sheaf (to find the biggest wall). Suppose $F$ is a subobject of $E$
in $\mathcal{B}^{\beta}$ with $\nu_{\beta}(F)=\nu_{\beta}(E)$ and
$\operatorname{ch}(F)=(r,x,y/2,z/6)$. Then $r\geq 1$ and $x\leq 0$. By [20,
Corollary 2.8], every $\nu$-wall must cross
$\beta=\mu(E)-\sqrt{\Delta(E)/r(E)^{2}}=-1$ (at the bottom of $\Theta_{v}$)
and so $0<x+r<2$ and then $x=1-r$. We look for $\nu$-walls along
$\Theta^{-}_{v}$. Then $\alpha^{2}=\beta^{2}-1$. Now
$\operatorname{ch}_{2}^{\alpha,\beta}(F)=y/2-\beta(1-r)+r/2=0$. So
$y=2\beta(1-r)-r$. If $r\geq 2$ then $y\geq r-1$ as $\beta<-1$. From Bogomolov
for $F$ we have $(r-1)^{2}\geq ry\geq r(r-1)$ which is impossible. So $r=1$.
Then $y=-1$ but $x=0$ and $(1,0,-1/2,z/6)$ is not the Chern character of a
sheaf.

Now we assume $\beta>0$. Then $E$ is the dual of a Gieseker semistable sheaf.
Suppose $0\to F\to E\to G\to 0$ is a short exact sequence in
$\mathcal{B}^{\beta}$ with
$\nu_{\alpha,\beta}(F)=\nu_{\alpha,\beta}(E)=\nu_{\alpha,\beta}(G)$ for some
$\alpha$. If $E$ is a sheaf then so is $G$, and
$G^{\vee}\in\mathcal{B}^{\beta}$ with
$\nu_{\alpha,-\beta}(G^{\vee})=-\nu_{\alpha,-\beta}(G)$. But then $F^{\vee}$
is also a sheaf and so $E^{\vee}$ is not $\nu_{\alpha,-\beta}$-stable but this
contradicts the last paragraph. If $E$ is not a sheaf then
$\mathcal{H}^{0}(G)$ is a quotient of $\mathcal{O}_{L}(1)$ and since we may
assume $G$ is $\nu_{\alpha,\beta}$-semistable, it follows condition (6) of
Proposition 2.15 holds and $\mathcal{H}^{-1}(G)$ is reflexive (any torsion
sheaf $T$ with a map $T[1]\to\mathcal{H}^{-1}(G)\to G$ injects). Since $S=0$
in Proposition 2.15, it follows that $G$ is the dual of a torsion free sheaf
and again $F^{\vee}$ is a sheaf. So we still have a contradiction. ∎

The previous lemma has two interesting applications. First we recover the
following stronger version of a well known fact.

######  Proposition 8.6.

If $k>0$ is an integer then there are no $\mu_{\leq 2}$-semistable torsion
free sheaves of Chern character $(2,0,-1,k)$ on ${\mathbb{P}^{3}}$.

###### Proof.

Note that $q(v)=8(2-9k^{2})<0$. Then if there are $\mu_{\leq 2}$-semistable
torsion free sheaves there must exist an actual vanishing $\nu$-wall by
Theorem 4.22. But there are no $\nu$-walls. ∎

Note that $S_{L}$ in the proof of Proposition 8.1 is an example of a
$\mu$-semistable reflexive sheaf of Chern character $(2,0,-1,1)$. In addition,
let $E$ be a null correlation bundle, and let $\\{p_{1},\dots,p_{k}\\}$ be
distinct points in ${\mathbb{P}^{3}}$; the kernel of an epimorphism
$E\twoheadrightarrow\oplus_{i=1}^{k}\mathcal{O}_{p_{i}}$ provides an example
of a $\mu$-stable torsion free sheaf with Chern character $(2,0,-1,-k)$ for
any $k>0$.

One can also provide a complete description of $\nu_{\alpha,\beta}$-stable
objects with Chern character $(2,0,-1,0)$.

######  Proposition 8.7.

Given any $\alpha>0$, an object $B\in\mathcal{B}^{\beta}({\mathbb{P}^{3}})$
with $\operatorname{ch}(B)=(2,0,-1,0)$ is $\nu_{\alpha,\beta}$-stable if and
only if

  1. (1)

$B$ is a null correlation sheaf, when $\beta<0$;

  2. (2)

$B^{\vee}[-1]$ is a null correlation sheaf, when $\beta>0$.

###### Proof.

Theorem 5.2 and Proposition 8.1 tell us that the asymptotically
$\nu_{\alpha,\beta}$-stable objects on each side of the $\alpha$-axis are
precisely the ones described above. Since there are no actual $\nu$-walls,
such objects are $\nu_{\alpha,\beta}$-stable everywhere. ∎

###  8.4. $\lambda$-walls and stability for the Chern character
$\mathbf{(2,0,-1,0)}$

We now turn to the classification of actual $\lambda$-walls corresponding to
Gieseker stable objects and the description of the
$\lambda_{\alpha,\beta,s}$-semistable objects with Chern character
$v=(2,0,-1,0)$ near the curve $\Gamma^{-}_{v,s}$.

Note that the fact that both $I_{L}$ and $I_{p}$ are
$\lambda_{1/\sqrt{6s+1},-2,s}$-semistable for every $s>0$, checked in Sections
8.1 and 8.2, imply that the sheaves of type (2.1) and (2.2) are also
$\lambda_{1/\sqrt{6s+1},-2,s}$-semistable for every $s>0$. For this case we
will set $s=1/3$ for simplicity.

We first observe that there is a vanishing wall through the point $P$ given by
$\beta=-2$ and $\alpha^{2}=1/3$ on $\Gamma^{-}_{v,1/3}$ given by
$F={\mathcal{O}}_{\mathbb{P}^{3}}(-1)$. Theorem 7.8 and Lemma 8.5 then imply
that this is the largest actual $\lambda$-wall for such Gieseker stable
sheaves and it suffices to find all of the walls through $P$. So in what
follows, we set $\beta=-2$ and $\alpha^{2}=1/3$. One strategy to find all of
the walls would be to guess some based on prior knowledge and then the
semistability of $F$ and $G$ would guarantee that the wall is an actual one.
But we shall try to deduce them by solving the inequalities to locate
solutions directly to illustrate how the techniques one might use to do this.

For $E$ with $\operatorname{ch}(E)=(2,0,-1,0)$ we consider a short exact
sequence in $\mathcal{A}^{\alpha,\beta}$

$0\to F\to E\to G\to 0$

with $\lambda_{\alpha,\beta}(F)=\lambda_{\alpha,\beta}(E)$ along
$\Gamma^{-}_{v,s}$. We let $\operatorname{ch}(F)=(r,x,y/2,z/6)$ for integers
$r,x,y,z$. Since $\chi(F)=r+11x/6+y+z/6$ is an integer, we must have
$6|(z-x)$. In fact, at our point $P$,
$\chi(F)\propto\operatorname{ch}_{3}^{P}(u)=0$ so this holds automatically.

We have $\operatorname{ch}^{\alpha,\beta}_{2}(E)=\frac{2}{3}\beta^{2}$,

$\operatorname{ch}^{\alpha,\beta}_{2}(F)=\frac{y}{2}-\beta
x+\frac{1}{3}\beta^{2}r+\frac{1}{2}r$

and

$\operatorname{ch}^{\alpha,\beta}_{2}(G)=-\frac{y}{2}+\beta
x-\frac{1}{3}\beta^{2}(r-2)-\frac{1}{2}r$

These provide the constraints:

(59) $0<\frac{y}{2}+2x+\frac{11}{6}r<\frac{8}{3}$

Since $F\in\mathcal{T}_{\mathcal{B}}$ we have $x\geq-2r$. Finally, we have
that $\lambda_{1/\sqrt{3},-2,1/3}(F)=0$ which gives

$z=-11x-6(r+y).$

Finally we have

(60)
$\begin{split}Q_{P}(F)&=12r^{2}+28rx+\frac{23}{3}ry+\frac{46}{3}x^{2}+8xy+y^{2}\qquad\text{and}\\\
Q_{P}(v-u)&=12r^{2}+28rx+\frac{23}{3}ry-\frac{98}{3}r+\frac{46}{3}x^{2}+8xy-40x+y^{2}-\frac{34}{3}y+\frac{64}{3}\end{split}$

From Proposition 2.12, there are some constraints on $F$ and $G$. We will
focus on cases (1) and (2). We first eliminate case (2).

######  Lemma 8.8.

Given $E$, $F$ and $G$ as above then $F\not\in\mathcal{C}oh(X)$ and
$G\not\in\mathcal{B}^{\beta}[1]\cap\mathcal{C}oh(X)[1]$.

###### Proof.

The assumption $G[-1]\in\mathcal{B}^{\beta}$ implies that $x\geq 2(r-2)$, and
$r\geq 2$ and so the constraints from $\operatorname{ch}_{2}^{P}(F)$ imply
that $y/2<32/3-35r/6\leq-1$ and so $y\leq-3$. We also have

$2x<\frac{8}{3}-\frac{y}{2}-\frac{11}{6}r.$

Then we have from (60),

$Q_{P}(F)<\frac{1601}{12}r^{2}-\frac{3440}{9}r+\frac{7648}{27}-3\left(y-\frac{32}{9}+\frac{7}{6}r\right)^{2}$

Now $y<64/3-35r/3$ and when $r>3$, it is easy to see that the above bound on
$Q_{P}(F)$ is negative. When $r=2$, we have $-22<3y+12x<-6$ from the
constrains while $Q_{P}(u)<0$ in the range
$-23-\sqrt{f(x)}<3y+12x<-23+\sqrt{f(x)}$, for some (quadratic) function $f(x)$
and $Q_{P}(v-u)<0$ in the range $-6-\sqrt{6}x<3y+12x<-6+\sqrt{6}x$ hence one
of these must be negative so long as the ranges overlap. This happens when
$x\geq 2$. When $x=0$, we have $G[-1]\in\mathcal{A}^{\alpha,\beta}$ which is a
contradiction. If $x=1$, then the constraints give $-17/3<y/2<-2$. But $G$ is
a sheaf and so $y\cong 0\pmod{2}$. Hence $y=-6,-8,-10$ are the only
possibilities. For $y=-6$, $Q_{P}(v-u)=-2/3$. For $y=-8$, $Q_{P}(u)=-10/3$.
For $y=-10$, $Q_{P}(u)=-14$. When $r=3$ the constraints give $-66<6y+24x<-34$
while $Q_{P}(u)<0$ in the range $-69-\sqrt{g(x)}<6y+24x<-69+\sqrt{g(x)}$ and
$Q_{P}(v-u)<0$ in the range $-35-\sqrt{h(x)}<6y+24x<-35+\sqrt{h(x)}$ and again
one or other must be negative for any value of $y$ or $x$ and this time the
ranges overlap for all $x\geq 0$.

In any case, $F$ or $G$ is not $\lambda_{P}$-semistable which is impossible. ∎

Combining this with Proposition 8.5, we deduce that
$\Delta_{21}(\operatorname{ch}(E),\operatorname{ch}(F))\geq 0$ (with equality
only if it holds for all $\beta$) which gives the inequality

(61) $y-\frac{4}{3}\beta x+r\leq 0.$

Another useful consequence of Proposition 8.5 is that
$Y_{(2,0,-1,0)}=\emptyset$ in Proposition 4.3 and the above Lemma together
with Proposition 7.12 then provide us with a useful necessary condition on $u$
since the numerical $\lambda$-wall cannot cross $\Theta^{-}_{v}$ by
Proposition 7.12.

######  Lemma 8.9.

For $v=(2,0,-1,0)$ if $u$ is the Chern character corresponding to an actual
$\lambda$-wall in $R^{L}_{v,s}$ satisfying $x\neq 0$ then $|y+r|\leq 2|x|$.

######  Proposition 8.10.

Consider $v=(2,0,-1,0)$. Then on $X={\mathbb{P}^{3}}$ there are exactly four
pseudo $\lambda$-walls for a Gieseker stable sheaf $E$ intersecting
$\Gamma^{-}_{v,1/3}$ given by

(62) $\displaystyle 0\to\mathcal{I}_{L}\to\hbox{}$ $\displaystyle
E\to\mathcal{I}_{p}\to 0$ (63) $\displaystyle 0\to K\to\hbox{}$ $\displaystyle
E\to\mathcal{O}_{L}(-1)\to 0$ (64) $\displaystyle
0\to\mathcal{O}(-1)\to\hbox{}$ $\displaystyle E\to\mathcal{I}_{C}(1)\to 0$
(65) $\displaystyle 0\to S_{p}(1)\to\hbox{}$ $\displaystyle
E\to\mathcal{I}_{p/H}\to 0,$

where $L$ is a line, $C$ is the union of two lines on a quadric surface in
${\mathbb{P}^{3}}$ and $H$ is a hyperplane; moreover, $K$ is a Gieseker
semistable sheaf with $\operatorname{ch}(K)=(2,0,-2,2)$ as described in
display (57). The first two are coincident and the first three are actual
$\lambda$-walls below $\Gamma^{-}_{v,1/3}$ while only (64) is actual above
$\Gamma^{-}_{v,1/3}$.

###### Proof.

We assume $E$ is a Gieseker stable torsion free sheaf. In fact, we shall only
assume the weaker condition that $E$ is a $\mu$-semistable torsion free sheaf.
These are classified by Proposition 8.1. Once we have found the walls for
these we need to also consider new objects which become $\lambda$-stable as we
cross each wall. Now our destabilizing objects
$F,G\in\mathcal{A}^{\alpha,\beta}\cap\mathcal{B}^{\beta}\cap\mathcal{C}oh(X)$.
Then we have additional constraints to add to (59) and (60):

(66) $-2r<x\leq 0\\\ $

since $E$ is $\mu$-semistable and $F\in\mathcal{B}^{\beta}$.

Consider the case $r=1$ first.

### Case $(1,0,y/2,z/6)$

Suppose $x=0$. Then the $\operatorname{ch}_{2}^{P}$ constraint gives
$5/3>y>-11/3$. Then $1\geq y\geq-3$. From Lemma 8.9 we have $y\leq-1$ and so
we have the possibilities $y=-1,-2,-3$. But $F$ is rank $1$ and torsion-free
and so $y\cong 0\pmod{2}$ and so $y=-2$ is the only possibility. This gives
the first wall (62).

### Case $(1,-1,y/2,z/6)$

Now suppose $x=-1$. The constraints on $y$ gives $17/6>y\geq 1$. But Lemma 8.9
implies $y<2-r=1$ and so $y=1$ is the only possibility and this gives wall
(64).

### Case $(2,x,y/2,z/6)$

Finally we return to the case $r=2$. Then $G$ is a torsion-sheaf. We have
$-3\leq x\leq 0$. We shall consider these, case by case as before.

### Case $(2,0,y/2,z/6)$

We now consider the special case $x=0$. Then $y<-2$ as
$\operatorname{ch}_{2}(G)>0$.Then $-6\leq y<-2$ But $y$ must be even for $G$
to be a sheaf and so $y\leq-4$. When $y=-4$ we have the same wall as (62) but
given by the sequence (63). When $y=-2$ then
$\operatorname{ch}(G)=\operatorname{ch}(E)$ which is impossible.

### Case $(2,-1,y/2,z/6)$

Now we let $x=-1$. Then the constraints give $0\geq y>-10/3$ and then $-3\leq
y\leq 0$. But $2\operatorname{ch}_{2}(G)\cong 1\pmod{2}$. Hence $y=-1$ or
$y=-3$. In the former case, $\operatorname{ch}(F)=(2,-1,-1/2,5/6)$ and this
corresponds to sequence (65) But one can check that
$Q_{P}(2,-1,-1/2,5/6)=1>0$, $Q_{P}(0,1,-1/2,-5/6)=25/3>0$ and
$Q_{P}(v)-Q_{P}(2,-1,-1/2,5/6)-Q_{P}(0,1,-1/2,-5/6)=12>0$ and so this is a
pseudo-wall. In the latter case, $Q_{P}(u)=-17/3$ and so $F$ cannot be
$\lambda$-semistable.

### Case $(2,{\leq-2},y/2,z/6)$

When $x=-2$, we have $1\leq y\leq 5$ from the $\operatorname{ch}_{2}^{P}$
constraints and $y+2\leq 4$ from Lemma 8.9. So $1\leq y\leq 2$. But
$2\operatorname{ch}_{2}(G)\cong 0\pmod{2}$ and so $y=2$ is the only possible
solution. Then $F={\mathcal{O}}_{\mathbb{P}^{3}}(-1)^{\oplus 2}$ and
$G={\mathcal{I}}_{C}(1)/{\mathcal{O}}_{\mathbb{P}^{3}}(-1)$ but this gives the
same wall as (64).

When $x=-3$, the $\operatorname{ch}_{2}^{P}$ constraint gives $y>14/3$ and os
$y\geq 5$. But Lemma 8.9, gives $y+2<6$ which is a contradiction.

This completes all of the possible numerical cases.

Note that the wall for (62) and (63) coincide. For $s\geq 1/3$ the walls below
$\Gamma^{-}_{v,s}$ are in the order (62)=(63)$>$(65)$>$(64) and this is
reversed above $\Gamma^{-}_{v,s}$ (see Figure 11).

We have now shown that the only walls which arise and which destabilize
$\mu$-semistable torsion free sheaves are the ones listed in Proposition 8.10.
But we still need to show that there are no further walls destabilizing new
objects which can arise as we cross one of these walls. This is because on
this wall and $E$ arising in (62) has a Jordan–Hölder cofiltration
$E\to\mathcal{I}_{p}\to\mathcal{O}_{L}(-1)\to 0$. This happens for the non
locally free null correlation sheaves. But non-trivial extensions of the form

$0\to\mathcal{I}_{p}\to F\to\mathcal{I}_{L}\to 0,$

which do exist for all points $p$ and lines $L$, are unstable below this wall
because there is an injection $\mathcal{I}_{L^{\prime}}\to\mathcal{I}_{p}$ in
$\mathcal{A}^{\alpha,\beta}$ for any $L^{\prime}\ni p$. On the other hand, if
$E$ is a Gieseker semistable sheaf then there is a sequence of the form (63).
For these, $\operatorname{Ext}^{1}(\mathcal{O}_{L}(-1),K)\neq 0$ and so there
are $\lambda_{\alpha,\beta,s}$-stable objects given as (flipped) non-trivial
extensions

(67) $0\to\mathcal{O}_{L}(-1)\to\widetilde{E}\to K\to 0.$

Note that $K$ sits in an extension

$0\to\mathcal{I}_{L}\to K\to\mathcal{I}_{p}\to 0$

and so $K$ is $\lambda$-semistable. So below this biggest wall and above the
next wall the only $\lambda$-stable objects are sheaves with torsion of the
form $F$ above. These cannot appear in extensions of the form (65). Indeed,
such extensions do exist and since such $E$ are locally-free Gieseker stable
they must also occur in sequences of the form (63) and so either $S_{p}(1)$ or
$\mathcal{I}_{p/H}$ must be $\lambda_{\alpha,\beta,s}$-unstable at that point
(in fact, it is $\mathcal{I}_{p/H}$ which is unstable). This shows that the
wall (65) is not an actual $\lambda$-wall. Finally, every object occurring as
kernels or quotients of the four pseudo $\lambda$-walls admit a map from
${\mathcal{O}}_{\mathbb{P}^{3}}(-1)$ and so (64) is a vanishing wall. Above
$\Gamma^{-}_{v,1/3}$ this is the largest wall and so it is the only actual
$\lambda$-wall above $\Gamma^{-}_{v,1/3}$ and below $\Theta_{v}^{-}$. ∎

######  Remark 8.11.

Note that the problem of eliminating walls when $x\leq-2$ and $r=1$ is
actually a quadratic programming problem. To see this note that

$z=-2\beta^{2}x+3\beta y+3\beta-3x$

and so

$Q_{P}(F)=3\beta^{2}+\frac{10}{3}\beta^{2}x^{2}+2x^{2}-2\beta^{3}x-6\beta
x-4\beta xy+y^{2}+\frac{5}{3}\beta^{2}y+y,$

which we can rewrite as

$F_{t}(\mathbf{v})=\mathbf{v}A_{t}\mathbf{v}^{T}-\mathbf{c}\mathbf{v}^{T}$

subject to the inequalities

$\displaystyle 4\leq\gamma<\lambda/2<\mu/4,$ $\displaystyle-2-\gamma\leq
y\leq\gamma,$
$\displaystyle\lambda-\frac{1}{3}\mu<\frac{y+1}{2}<\lambda+\frac{1}{3}\mu,$

where $\mathbf{v}=(\lambda,\mu,\gamma,y)=(\beta x,\beta^{2},x^{2},y)$ and

$A_{t}=\begin{pmatrix}10(1-t)/3&-1&0&-2\\\ -1&0&5t/3&5/6\\\ 0&5t/3&0&0\\\
-2&5/6&0&1\end{pmatrix}$

and $c=(6,-3,-2,-1)$. Then we need to show there is a real value of $t$ for
which $F_{t}(\mathbf{v})<0$ given the constraints. In fact, numerical methods
show that the ”local” maximum value of $F(\mathbf{v})$ when $t=1$ is negative
even without the additional non-linear constraint $\gamma\mu=\lambda^{2}$. But
$A_{t}$ is not definite and so this need not be a global maximum.

Figure 11. This graph shows the actual and pseudo $\lambda$-walls for the
Chern character $v=(2,0,-1,0)$ intersecting at the point
$(\alpha=1/\sqrt{3},\beta=-2)$, as described in Proposition 8.10. The black
and purple curves are actual $\lambda$-walls, while the gren curve is only a
pseudo $\lambda$-wall. The stability chambers in the region $R^{-}_{v}$ are
described in Theorem 8.12.

It follows from Theorem 6.17 that no other $\lambda$-walls in the region
$R^{-}_{v}$ exist, when $s=1/3$. This observation allows us to give a complete
chamber decomposition of this region, summarized in the following statement.

######  Theorem 8.12.

Let $X={\mathbb{P}^{3}}$ and consider the numerical Chern character
$v=(2,0,-1,0)$; fix $s=1/3$.

The region $R^{-}_{v}$ is divided into three stability chambers $C_{i}$ within
which the $\lambda_{\alpha,\beta,s}$-stable objects are described as follows:

  * ($C_{1}$)

null correlation sheaves;

  * ($C_{2}$)

nontrivial extensions of a semistable torsion free sheaf $K$ with
$\operatorname{ch}(K)=(2,0,-2,2)$ by $\mathcal{O}_{L}(-1)$, where $L$ is a
line;

  * ($C_{3}$)

no stable objects.

The three stability chambers just described are pictured in Figure 11 as
follows. The chamber (C1) lies above the purple and black curves (which
corresponds to the walls (62) and (64), respectively); the chamber (C2) is
located in the sector between the purple and the black curves below
$\Gamma_{v,1/3}^{-}$; finally, the vanishing chamber lies below the black
curve. The green curve corresponds to the pseudo $\lambda$-wall (65).

As a final remark, we observe that Proposition 8.10 provide concrete examples
of intersecting actual $\lambda$-walls for the Chern character $(2,0,-1,0)$.

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