content.tex

\section{Introduction}
\label{sec:intro}

The theory of Bridgeland stability conditions \cite{BridgelandTom2007SCoT} on
complexes of sheaves was developed as a generalisation of stability for vector
bundles. The definition is most analoguous to Mumford stability, but is more
aware of the features that sheaves can have on spaces of dimension greater
than 1. Whilst also asymptotically matching up with Gieseker stability.
For K3 surfaces, explicit stability conditions were defined in
\cite{Bridgeland_StabK3}, and later shown to also be valid on other surfaces.

The moduli spaces of stable objects of some fixed Chern character $v$ is
studied, as well as how they change as we vary the Bridgeland stability
condition. They in fact do not change over whole regions of the stability
space (called chambers), but do undergo changes as we cross `walls' in the
stability space. These are where there is some stable object $F$ of $v$ which
has a subobject who's slope overtakes the slope of $v$, making $F$ unstable
after crossing the wall.

% NOTE: SURFACE SPECIALIZATION
% (come back to these when adjusting to general Picard rank 1)
In this document we concentrate on two surfaces: Principally polarized abelian
surfaces and the projective surface $\PP^2$. Although this can be generalised
for Picard rank 1 surfaces, the formulae will need adjusting.
The Bridgeland stability conditions (defined in \cite{Bridgeland_StabK3}) are
given by two parameters $\alpha \in \RR_{>0}$, $\beta \in \RR$, which will be
illustrated throughout this article with diagrams of the upper half plane.

It is well known that for any rational $\beta_0$,
the vertical line $\{\sigma_{\alpha,\beta_0} \colon \alpha \in \RR_{>0}\}$ only
intersects finitely many walls
\cite[Thm 1.1]{LoJason2014Mfbs}
\cite[Prop 4.2]{alma9924569879402466}
\cite[Lemma 5.20]{MinaHiroYana_SomeModSp}.
A consequence of this is that if
$\beta_{-}$ is rational, then there can only be finitely many circular walls to the
left of the vertical wall $\beta = \mu$.
On the other hand, when $\beta_{-}$ is not rational, \cite{yanagida2014bridgeland}
showed that there are infinitely many walls.

This dichotomy does not only hold for real walls, realised by actual objects in
$\bddderived(X)$, but also for pseudowalls. Here pseudowalls are defined as
`potential' walls, induced by hypothetical Chern characters of semistabilizers
which satisfy certain numerical conditions which would be satisfied by any real
destabilizer, regardless of whether they are realised by actual semistabilizers
in $\bddderived(X)$ (dfn \ref{dfn:pseudo-semistabilizer}).

Since real walls are a subset of pseudowalls, the irrational $\beta_{-}$ case
follows immediately from the corresponding case for real walls.
However, the rational $\beta_{-}$ case involves showing that the following
conditions only admit finitely many solutions (despite the fact that the same
conditions admit infinitely many solutions when $\beta_{-}$ is irrational).


For a semistabilizing sequence
$E \hookrightarrow F \twoheadrightarrow G$ in $\mathcal{B}^\beta$
we have the following conditions.
There are some Bogomolov-Gieseker inequalities:
$0 \leq \Delta(E), \Delta(G)$.
We also have a condition relating to the tilt category $\firsttilt\beta$:
$0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)$.
Finally, there is a condition ensuring that the radius of the circular wall is
strictly positive: $\chern^{\beta_{-}}_2(E) > 0$.

For any fixed $\chern_0(E)$, the inequality
$0 \leq \chern^{\beta}_1(E) \leq \chern^{\beta}_1(F)$,
allows us to bound $\chern_1(E)$. Then, the other inequalities allow us to
bound $\chern_2(E)$. The final part to showing the finiteness of pseudowalls
would be bounding $\chern_0(E)$. This has been hinted at in
\cite{SchmidtBenjamin2020Bsot} and done explicitly by Benjamin Schmidt within a
SageMath \cite{sagemath} library which computes pseudowalls
\cite{SchmidtGithub2020}.
Here we discuss these bounds in more detail, along with the methods used,
followed by refinements on them which give explicit formulae for tighter bounds
on $\chern_0(E)$ of potential destabilizers $E$ of $F$.


\section{Setting and Definitions: Clarifying `pseudo'}

%\begin{definition}[Twisted Chern Character]
%\label{sec:twisted-chern}
%For a given $\beta$, define the twisted Chern character as follows.
%\[\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)\]
%\noindent
%Component-wise, this is:
%\begin{align*}
%	\chern^\beta_0(E) &= \chern_0(E)
%\\
%	\chern^\beta_1(E) &= \chern_1(E) - \beta \chern_0(E)
%\\
%	\chern^\beta_2(E) &= \chern_2(E) - \beta \chern_1(E) + \frac{\beta^2}{2} \chern_0(E)
%\end{align*}
%where $\chern_i$ is the coefficient of $\ell^i$ in $\chern$.
%
%% TODO I think this^ needs adjusting for general Surface with $\ell$
%\end{definition}
%
%$\chern^\beta_1(E)$ is the imaginary component of the central charge
%$\centralcharge_{\alpha,\beta}(E)$ and any element of $\firsttilt\beta$
%satisfies $\chern^\beta_1 \geq 0$.

Throughout this article, as noted in the introduction, we will be exclusively
working over surfaces $X$ with Picard rank 1, with a choice of ample line bundle
$L$ such that $\ell\coloneqq c_1(L)$ generates $NS(X)$.
We take $m\coloneqq \ell^2$ as this will be the main quantity which will
affect the results.

\begin{definition}[Pseudo-semistabilizers]
\label{dfn:pseudo-semistabilizer}
% NOTE: SURFACE SPECIALIZATION
	Given a Chern Character $v$, and a given stability
	condition $\sigma_{\alpha,\beta}$,
	a \textit{pseudo-semistabilizing} $u$ is a `potential' Chern character:
	\[
		u = \left(r, c\ell, \frac{e}{\lcm(m,2)} \ell^2\right)
		\qquad
		r,c,e \in \ZZ
	\]
	which has the same tilt slope as $v$:
	$\nu_{\alpha,\beta}(u) = \nu_{\alpha,\beta}(v)$.

	\noindent
	Furthermore the following inequalities are satisfied:
	\begin{itemize}
		\item $\Delta(u) \geq 0$
		\item $\Delta(v-u) \geq 0$
		\item $0 \leq \chern_1^{\beta}(u) \leq \chern_1^{\beta}(v)$
	\end{itemize}

	Note $u$ does not need to be a Chern character of an actual sub-object of some
	object in the stability condition's heart with Chern character $v$.
\end{definition}

At this point, and in this document, we do not care about whether
pseudo-semistabilizers are even Chern characters of actual elements of
$\bddderived(X)$, some other sources may have this extra restriction too.

Later, Chern characters will be written $(r,c\ell,d\ell^2)$ because operations
(such as multiplication) are more easily defined in terms of the coefficients of
the $\ell^i$. However, at the end, it will become important again that
$d \in \frac{1}{\lcm(m,2)}\ZZ$.

\begin{definition}[Pseudo-walls]
\label{dfn:pseudo-wall}
	Let $u$ be a pseudo-semistabilizer of $v$, for some stability condition.
	Then the \textit{pseudo-wall} associated to $u$ is the set of all stablity
	conditions where $u$ is a pseudo-semistabilizer of $v$.
\end{definition}

% TODO possibly reference forwards to Bertram's nested wall theorem section to 
% cover that being a pseudo-semistabilizer somewhere implies also on whole circle

\begin{lemma}[Sanity check for Pseudo-semistabilizers]
	Given a stability
	condition $\sigma_{\alpha,\beta}$,
	if $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing sequence in
	$\firsttilt\beta$ for $F$.
	Then $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$
\end{lemma}

\begin{proof}
	Suppose $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing
	sequence with respect to a stability condition $\sigma_{\alpha,\beta}$.
	\begin{equation*}
		\chern(E) = -\chern(\homol^{-1}_{\coh}(E)) + \chern(\homol^{0}_{\coh}(E))
	\end{equation*}
	Therefore, $\chern(E)$ is of the form
	$(r,c\ell,\frac{e}{\lcm(m,2)}\ell^2)$
	provided that this is true for any coherent sheaf.
	For any coherent sheaf $H$, we have the following:
	\begin{equation*}
		\chern(H) = \left(c_0(H), c_1(H), - c_2(H) + \frac{1}{2} {c_1(H)}^2\right)
	\end{equation*}
	Given that $\ell$ generates the Neron-Severi group, $c_1(H)$ can then be
	written $c\ell$.
	\begin{equation*}
		\chern(H) = \left(
			c_0(H), c\ell,
			\left(- \frac{c_2(H)}{\ell^2} + \frac{c^2}{2} \right)\ell^2
		\right)
	\end{equation*}
	This fact along with $c_0$, $c_2$ being an integers on surfaces, and
	$m\coloneqq \ell^2$ implies that $\chern(H)$
	(hence $\chern(E)$ too) is of the required form.
	

	Since all the objects in the sequence are in $\firsttilt\beta$, we have
	$\chern_1^{\beta} \geq 0$ for each of them. Due to additivity
	($\chern(F) = \chern(E) + \chern(G)$), we can deduce
	$0 \leq \chern_1^{\beta}(E) \leq \chern_1^{\beta}(F)$.


	$E \hookrightarrow F \twoheadrightarrow G$ being a semistabilizing sequence
	means	$\nu_{\alpha,\beta}(E) = \nu_{\alpha,\beta}(F) = \nu_{\alpha,\beta}(F)$.
	% MAYBE: justify this harder
	But also, that this is an instance of $F$ being semistable, so $E$ must also
	be semistable
	(otherwise the destabilizing subobject would also destabilize $F$).
	Similarly $G$ must also be semistable too.
	$E$ and $G$ being semistable implies they also satisfy the Bogomolov
	inequalities:
	% TODO ref Bogomolov inequalities for tilt stability
	$\Delta(E), \Delta(G) \geq 0$.
	Expressing this in terms of Chern characters for $E$ and $F$ gives:
	$\Delta(\chern(E)) \geq 0$ and $\Delta(\chern(F)-\chern(E)) \geq 0$.

\end{proof}


\section{Characteristic Curves of Stability Conditions Associated to Chern
Characters}

% NOTE: SURFACE SPECIALIZATION
Considering the stability conditions with two parameters $\alpha, \beta$ on
Picard rank 1 surfaces.
We can draw 2 characteristic curves for any given Chern character $v$ with
$\Delta(v) \geq 0$ and positive rank.
These are given by the equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$.

\begin{definition}[Characteristic Curves $V_v$ and $\Theta_v$]
Given a Chern character $v$, with positive rank and $\Delta(v) \geq 0$, we
define two characteristic curves on the $(\alpha, \beta)$-plane:

\begin{align*}
	V_v &\colon \:\: \chern_1^{\alpha, \beta}(v) = 0 \\
	\Theta_v &\colon \:\: \chern_2^{\alpha, \beta}(v) = 0
\end{align*}
\end{definition}

\subsection{Geometry of the Characteristic Curves}

These characteristic curves for a Chern character $v$ with $\Delta(v)\geq0$ are
not affected by flipping the sign of $v$ so it's only necessary to consider
non-negative rank.
As discussed in subsection \ref{subsect:relevance-of-V_v}, making this choice
has Gieseker stable coherent sheaves appearing in the heart of the stability
condition $\firsttilt{\beta}$ as we move `left' (decreasing $\beta$).

\subsubsection{Positive Rank Case}
\label{subsect:positive-rank-case-charact-curves}

\begin{fact}[Geometry of Characteristic Curves in Positive Rank Case]
The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
as well as the restrictions on $v$, when $\chern_0(v)>0$:
\begin{itemize}
	\item $V_v$ is a vertical line at $\beta=\mu(v)$
	\item $\Theta_v$ is a hyperbola with assymptotes angled at $\pm 45^\circ$
		crossing where $V_v$ meets the $\beta$-axis: $(\mu(v),0)$
	\item $\Theta_v$ is oriented with left-right branches (as opposed to up-down).
		The left branch shall be labelled $\Theta_v^-$ and the right $\Theta_v^+$.
	\item The gap along the $\beta$-axis between either branch of $\Theta_v$
		and $V_v$ is $\sqrt{\Delta(v)}/\chern_0(v)$.
	\item When $\Delta(v)=0$, $\Theta_v$ degenerates into a pair of lines, but the
		labels $\Theta_v^\pm$ will still be used for convenience.
\end{itemize}
\end{fact}

These are illustrated in Fig \ref{fig:charact_curves_vis}
(dotted line for $i=1$, solid for $i=2$).

\begin{sagesilent}
from characteristic_curves import \
typical_characteristic_curves, \
degenerate_characteristic_curves
\end{sagesilent}


\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
	\centering
	\sageplot[width=\textwidth]{typical_characteristic_curves}
	\caption{$\Delta(v)>0$}
	\label{fig:charact_curves_vis_bgmvlPos}
\end{subfigure}%
\hfill
\begin{subfigure}{.49\textwidth}
	\centering
	\sageplot[width=\textwidth]{degenerate_characteristic_curves}
	\caption{
		$\Delta(v)=0$: hyperbola collapses
	}
	\label{fig:charact_curves_vis_bgmlv0}
\end{subfigure}
\caption{
	Characteristic curves ($\chern_i^{\alpha,\beta}(v)=0$) of stability conditions
	associated to Chern characters $v$ with $\Delta(v) \geq 0$ and positive rank.
}
\label{fig:charact_curves_vis}
\end{figure}

\begin{definition}[$\beta_{\pm}$]
	\label{dfn:beta_pm}
	Given a formal Chern character $v$ with positive rank, we define $\beta_{\pm}(v)$ to be
	the $\beta$-coordinate of where $\Theta_v^{\pm}$ meets the $\beta$-axis:
	\[
		\beta_\pm(R,C\ell,D\ell^2) = \frac{C \pm \sqrt{C^2-2RD}}{R}
	\]
	\noindent
	In particular, this means $\beta_\pm(v)$ are the two roots of the quadratic
	equation $\chern_2^{\beta}(v)=0$.

	This definition will be extended to the rank 0 case in definition \ref{dfn:beta_-_rank0}.
\end{definition}


\subsubsection{Rank Zero Case}
\label{subsubsect:rank-zero-case-charact-curves}

\begin{sagesilent}
from rank_zero_case import Theta_v_plot
\end{sagesilent}

\begin{fact}[Geometry of Characteristic Curves in Rank 0 Case]
The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
as well as the restrictions on $v$, when $\chern_0(v)=0$ and $\chern_1(v)>0$:


\begin{minipage}{0.5\textwidth}
\begin{itemize}
	\item $V_v = \emptyset$
	\item $\Theta_v$ is a vertical line at $\beta=\frac{D}{C}$
		where $v=\left(0,C\ell,D\ell^2\right)$
\end{itemize}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
	\sageplot[width=\textwidth]{Theta_v_plot}
	%\caption{$\Delta(v)>0$}
	%\label{fig:charact_curves_rank0}
\end{minipage}
\end{fact}

We can view the characteristic curves for $\left(0,C\ell, D\ell^2\right)$ with $C>0$ as
the limiting behaviour of those of $\left(\varepsilon, C\ell, D\ell^2\right)$.
Indeed:
\begin{align*}
	\mu\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C}{\varepsilon} &\longrightarrow +\infty
	\\
	\text{as} \:\: 0<\varepsilon &\longrightarrow 0
\end{align*}
So we can view $V_v$ as moving off infinitely to the right, with $\Theta_v^+$ even further.
But also, considering the base point of $\Theta_v^-$:
\begin{align*}
	\beta_{-}\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C - \sqrt{C^2-2D\varepsilon}}{\varepsilon}
	&\longrightarrow \frac{D}{C}
	\\
	\text{as} \:\: 0<\varepsilon &\longrightarrow 0
	&\text{(via L'H\^opital)}
\end{align*}

So we can view $\Theta_v^-$ as approaching the vertical line that $\Theta_v$  becomes.
For this reason, I will refer to the whole of $\Theta_v$ in the rank zero case
as $\Theta_v^-$ to be able to use the same terminology in both positive rank
and rank zero cases.

\begin{definition}[Extending $\beta_-$ to rank 0 case]
	\label{dfn:beta_-_rank0}
	Given a formal Chern character $v$ with rank 0 and $\chern_1(v)>0$, we define
	$\beta_-(v)$ to be the $\beta$-coordinate of point where $\Theta_v$ meets the
	$\beta$-axis:
	\[
		\beta_-(0,C\ell,D\ell^2) = \frac{D}{C}
	\]
	\noindent
	If $\beta_+$ were also to be generalised to the rank 0 case, we would consider
	its value to be $+\infty$ due to the discussion above.
\end{definition}


\subsection{Relevance of \texorpdfstring{$V_v$}{V_v}}
\label{subsect:relevance-of-V_v}

For the positive rank case, by definition of the first tilt $\firsttilt\beta$, objects of Chern character
$v$ can only be in $\firsttilt\beta$ on the left of $V_v$, where
$\chern_1^{\alpha,\beta}(v)>0$, and objects of Chern character $-v$ can only be
in $\firsttilt\beta$ on the right, where $\chern_1^{\alpha,\beta}(-v)>0$. In
fact, if there is a Gieseker semistable coherent sheaf $E$ of Chern character
$v$, then $E \in \firsttilt\beta$ if and only if $\beta<\mu(E)$ (left of the
$V_v$), and $E[1] \in \firsttilt\beta$ if and only if $\beta\geq\mu(E)$.
Because of this, when using these characteristic curves, only positive ranks are
considered, as negative rank objects are implicitly considered on the right hand
side of $V_v$.

In the rank zero case, this still applies if we consider $V_v$ to be
`infinitely to the right' ($\mu(v) = +\infty$). Precisely, Gieseker semistable
coherent sheaves $E$ of Chern character $v$ are contained in
$\firsttilt{\beta}$ for all $\beta$



\subsection{Relevance of \texorpdfstring{$\Theta_v$}{Θ_v}}

Since $\chern_2^{\alpha, \beta}$ is the numerator of the tilt slope
$\nu_{\alpha, \beta}$. The curve $\Theta_v$, where this is 0, firstly divides the
$(\alpha$-$\beta)$-half-plane into regions where the signs of tilt slopes of
objects of Chern character $v$ (or $-v$) are fixed. Secondly, it gives more of a
fixed target for some $u=(r,c\ell,d\frac{1}{2}\ell^2)$ to be a
pseudo-semistabilizer of $v$, in the following sense: If $(\alpha,\beta)$, is on
$\Theta_v$, then for any $u$, $u$ can only be a pseudo-semistabilizer of $v$ if
$\nu_{\alpha,\beta}(u)=0$, and hence $\chern_2^{\alpha, \beta}(u)=0$. In fact,
this allows us to use the characteristic curves of some $v$ and $u$ (with
$\Delta(v), \Delta(u)\geq 0$ and positive ranks) to determine the location of
the pseudo-wall where $u$ pseudo-semistabilizes $v$. This is done by finding the
intersection of $\Theta_v$ and $\Theta_u$, the point $(\beta,\alpha)$ where
$\nu_{\alpha,\beta}(u)=\nu_{\alpha,\beta}(v)=0$, and a pseudo-wall point on
$\Theta_v$, and hence the apex of the circular pseudo-wall with centre $(\beta,0)$
(as per subsection \ref{subsect:bertrams-nested-walls}).


\subsection{Bertram's Nested Wall Theorem}
\label{subsect:bertrams-nested-walls}

Although Bertram's nested wall theorem can be proved more directly, it's also
important for the content of this document to understand the connection with
these characteristic curves.
Emanuele Macri noticed in (TODO ref) that any circular wall of $v$ reaches a critical
point on $\Theta_v$ (TODO ref). This is a consequence of
$\frac{\delta}{\delta\beta} \chern_2^{\alpha,\beta} = -\chern_1^{\alpha,\beta}$.
This fact, along with the hindsight knowledge that non-vertical walls are
circles with centers on the $\beta$-axis, gives an alternative view to see that
the circular walls must be nested and non-intersecting.

\subsection{Characteristic Curves for Pseudo-semistabilizers}

These characteristic curves introduced are convenient tools to think about the
numerical conditions that can be used to test for pseudo-semistabilizers, and
for solutions to the problems
(\ref{problem:problem-statement-1},\ref{problem:problem-statement-2})
tackled in this article (to be introduced later).
In particular, problem (\ref{problem:problem-statement-1}) will be translated to
a list of numerical inequalities on it's solutions $u$.
% ref to appropriate lemma when it's written

The next lemma is a key to making this translation and revolves around the
geometry and configuration of the characteristic curves involved in a
semistabilizing sequence.

\begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
\label{lem:pseudo_wall_numerical_tests}
Let $v$ and $u$ be Chern characters with $\Delta(v),
\Delta(u)\geq 0$, and $v$ has non-negative rank (and $\chern_1(v)>0$ if rank 0).
Let $P$ be a point on $\Theta_v^-$.

\noindent
The following conditions:
\begin{enumerate}
\item $u$ is a pseudo-semistabilizer of $v$ at some point on $\Theta_v^-$ above
	$P$
\item $u$ destabilizes $v$ going `inwards', that is,
	$\nu_{\alpha,\beta}(u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
	$\nu_{\alpha,\beta}(u)>\nu_{\alpha,\beta}(v)$ inside.
\end{enumerate}

\noindent
are equivalent to the following more numerical conditions:
\begin{enumerate}
	\item $u$ has positive rank
	\item $\beta(P)<\mu(u)<\mu(v)$, i.e. $V_u$ is strictly between $P$ and $V_v$.
	\item $\chern_1^{\beta(P)}(u) \leq \chern_1^{\beta(P)}(v)$, $\Delta(v-u) \geq 0$
	\item $\chern_2^{P}(u)>0$
\end{enumerate}
\end{lemma}

\begin{proof}
Let $u,v$ be Chern characters with
$\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.


For the forwards implication, assume that the suppositions of the lemma are
satisfied. Let $Q$ be the point on $\Theta_v^-$ (above $P$) where $u$ is a
pseudo-semistabilizer of $v$.
Firstly, consequence 3 is part of the definition for $u$ being a
pseudo-semistabilizer at a point with same $\beta$ value of $P$ (since the
pseudo-wall surrounds $P$).
If $u$ were to have 0 rank, it's tilt slope would be decreasing as $\beta$
increases, contradicting supposition b. So $u$ must have strictly non-zero rank,
and we can consider it's characteristic curves (or that of $-u$ in case of
negative rank).
$\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too. This means that $\Theta_{\pm u}$ must
intersect $\Theta_v^-$ at $Q$. Considering the shapes of the hyperbolae alone,
there are 3 distinct ways that they can intersect, as illustrated in Fig
\ref{fig:hyperbol-intersection}. These cases are distinguished by whether it is
the left, or the right branch of $\Theta_u$ involved, as well as the positions
of the base. However, considering supposition b, only case 3 (green in
figure) is possible. This is because we need $\nu_{P}(u)>0$ (or $\nu_{P}(-u)>0$ in
case 1 involving $\Theta_u^+$), to satisfy supposition b.
Recalling how the sign of $\nu_{\alpha,\beta}(\pm u)$ changes (illustrated in
Fig \ref{fig:charact_curves_vis}), we can eliminate cases 1 and 2.

\begin{sagesilent}
from characteristic_curves import \
hyperbola_intersection_plot, \
correct_hyperbola_intersection_plot
\end{sagesilent}

\begin{figure}
\begin{subfigure}[t]{0.48\textwidth}
	\centering
	\sageplot[width=\textwidth]{hyperbola_intersection_plot()}
	\caption{Three ways the characteristic hyperbola for $u$ can intersect the left
	branch of the characteristic hyperbola for $v$}
	\label{fig:hyperbol-intersection}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\textwidth}
	\centering
	\sageplot[width=\textwidth]{correct_hyperbola_intersection_plot()}
	\caption{Closer look at characteristic curves for valid case}
	\label{fig:correct-hyperbol-intersection}
\end{subfigure}
\end{figure}

Fixing attention on the only possible case (2), illustrated in Fig
\ref{fig:correct-hyperbol-intersection}.
$P$ is on the left of $V_{\pm u}$ (first part of consequence 2), so $u$ must
have positive rank (consequence 1)
to ensure that $\chern_1^{\beta(P)} \geq 0$ (since the pseudo-wall passed over
$P$).
Furthermore, $P$ being on the left of $V_u$ implies
$\chern_1^{\beta{P}}(u) \geq 0$,
and therefore $\chern_2^{P}(u) > 0$ (consequence 4) to satisfy supposition b.
Next considering the way the hyperbolae intersect, we must have $\Theta_u^-$ taking a
base-point to the right $\Theta_v$, but then, further up, crossing over to the
left side. The latter fact implies that the assymptote for $\Theta_u^-$ must be
to the left of the one for $\Theta_v^-$. Given that they are parallel and
intersect the $\beta$-axis at $\beta=\mu(u)$ and $\beta=\mu(v)$ respectively. We
must have $\mu(u)<\mu(v)$ (second part of consequence 2),
that is, $V_u$ is strictly to the left of $V_v$.


Conversely, suppose that the consequences 1-4 are satisfied. Consequence 2
implies that the assymptote for $\Theta_u^-$ is to the left of $\Theta_v^-$.
Consequence 4, along with $\beta(P)<\mu(u)$, implies that $P$ must be in the
region left of $\Theta_u^-$. These two facts imply that $\Theta_u^-$ is to the
right of $\Theta_v^-$ at $\alpha=\alpha(P)$, but crosses to the left side as
$\alpha \to +\infty$, intersection at some point $Q$ above $P$.
This implies that the characteristic curves for $u$ and $v$ are in the
configuration illustrated in Fig \ref{fig:correct-hyperbol-intersection}.
We then have $\nu(u)=\nu(v)$ along a circle to the left of $V_u$ reaching it's
apex at $Q$, and encircling $P$. This along with consequence 3 implies that $u$
is a pseudo-semistabilizer at the point on the circle with $\beta=\beta(P)$.
Therefore, it's also a pseudo-semistabilizer further along the circle at $Q$
(supposition a).
Finally, consequence 4 along with $P$ being to the left of $V_u$ implies
$\nu_P(u) > 0$ giving supposition b.

The case with rank 0 can be handled the same way.

\end{proof}

\section{The Problem: Finding Pseudo-walls}

As hinted in the introduction (\ref{sec:intro}), the main motivation of the
results in this article are not only the bounds on pseudo-semistabilizer
ranks;
but also applications for finding a list (comprehensive or subset) of
pseudo-walls.

After introducing the characteristic curves of stability conditions associated
to a fixed Chern character $v$, we can now formally state the problems that we
are trying to solve for.

\subsection{Problem statements}

\begin{problem}[sufficiently large `left' pseudo-walls]
\label{problem:problem-statement-1}

Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
and $\Delta(v) \geq 0$.
The goal is to find all pseudo-semistabilizers $u$
which give circular pseudo-walls containing some fixed point
$P\in\Theta_v^-$.
With the added restriction that $u$ `destabilizes' $v$ moving `inwards', that is,
$\nu(u)>\nu(v)$ inside the circular pseudo-wall.
\end{problem}
This will give all pseudo-walls between the chamber corresponding to Gieseker
stability and the stability condition corresponding to $P$.
The purpose of the final `direction' condition is because, up to that condition,
semistabilizers are not distinguished from their corresponding quotients:
Suppose $E\hookrightarrow F\twoheadrightarrow G$, then the tilt slopes
$\nu_{\alpha,\beta}$
are strictly increasing, strictly decreasing, or equal across the short exact
sequence (consequence of the see-saw principle).
In this case, $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$, if and
only if $\chern(G)$ is a pseudo-semistabilizer of $\chern(F)$.
The numerical inequalities in the definition for pseudo-semistabilizer cannot
tell which of $E$ or $G$ is the subobject.
However, what can be distinguished is the direction across the wall that
$\chern(E)$ or $\chern(G)$ destabilizes $\chern(F)$
(they will each destabilize in the opposite direction to the other).
The `inwards' semistabilizers are preferred because we are moving from a
typically more familiar chamber
(the stable objects of Chern character $v$ in the outside chamber will only be
Gieseker stable sheaves).

Also note that this last restriction does not remove any pseudo-walls found,
and if we do want to recover `outwards' semistabilizers, we can simply take
$v-u$ for each solution $u$ of the problem.


\begin{problem}[all `left' pseudo-walls]
\label{problem:problem-statement-2}

Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
$\Delta(v) \geq 0$, and $\beta_{-}(v) \in \QQ$.
The goal is to find all pseudo-semistabilizers $u$ which give circular
pseudo-walls on the left side of $V_v$.
\end{problem}

This is a specialization of problem (\ref{problem:problem-statement-1})
with the choice $P=(\beta_{-},0)$, the point where $\Theta_v^-$ meets the
$\beta$-axis.
This is because all circular walls left of $V_v$ intersect $\Theta_v^-$ (once).
The $\beta_{-}(v) \in \QQ$ condition is to ensure that there are finitely many
solutions. As mentioned in the introduction (\ref{sec:intro}), this is known,
however this will also be proved again implicitly in section
\ref{sect:prob2-algorithm}, where an algorithm is produced to find all
solutions.

This description still holds for the case of rank 0 case if we consider $V_v$ to
be infinitely far to the right
(see section \ref{subsubsect:rank-zero-case-charact-curves}).
Note also that the condition on $\beta_-(v)$ always holds for $v$ rank 0.

\subsection{Numerical Formulations of the Problems}

The problems introduced in this section are phrased in the context of stability
conditions. However, these can be reduced down completely to purely numerical
problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.

\begin{lemma}[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
	\label{lem:num_test_prob1}
	Given a Chern character $v$ with non-negative rank
	(and $\chern_1(v)>0$ if rank 0),
	and $\Delta(v) \geq 0$,
	and a choice of point $P$ on $\Theta_v^-$.
	Solutions $u=(r,c\ell,d\ell^2)$
	to problem \ref{problem:problem-statement-1}.
	Are precisely given by $r,c \in \ZZ$, $d \in \frac{1}{\lcm(m,2)}$
	satisfying the following conditions:
	\begin{enumerate}
		\begin{multicols}{2}
		\item $r > 0$
			\label{item:rankpos:lem:num_test_prob1}
		\item $\Delta(u) \geq 0$
			\label{item:bgmlvu:lem:num_test_prob1}
		\item $\Delta(v-u) \geq 0$
			\label{item:bgmlvv-u:lem:num_test_prob1}
		\item $\mu(u)<\mu(v)$
		\item $0\leq\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$
			\label{item:chern1bound:lem:num_test_prob1}
		\item $\chern_2^{P}(u)>0$
			\label{item:radiuscond:lem:num_test_prob1}
		\end{multicols}
	\end{enumerate}
\end{lemma}

\begin{proof}
	Consider the context of $v$ being a Chern character with non-negative rank
	(and $\chern_1(v)>0$ if rank 0)
	and
	$\Delta \geq 0$, and $u$ being a Chern character with $\Delta(u) \geq 0$.
	Lemma \ref{lem:pseudo_wall_numerical_tests} gives that the remaining
	conditions for $u$ being a solution to problem
	\ref{problem:problem-statement-1} are precisely equivalent to the
	remaining conditions in this lemma.
	% TODO maybe make this more explicit
	% (the conditions are not exactly the same)

\end{proof}

\begin{corollary}[Numerical Tests for All `left' Pseudo-walls]
\label{cor:num_test_prob2}
	Given a Chern character $v$ with positive rank and $\Delta(v) \geq 0$,
	such that $\beta_{-}\coloneqq\beta_{-}(v) \in \QQ$.
	Solutions $u=(r,c\ell,d\ell^2)$
	to problem \ref{problem:problem-statement-2}.
	Are precisely given by $r,c \in \ZZ$, $d\in\frac{1}{\lcm(m,2)}\ZZ$ satisfying
	the following conditions:
	\begin{enumerate}
		\begin{multicols}{2}
		\item $r > 0$
			\label{item:rankpos:lem:num_test_prob2}
		\item $\Delta(u) \geq 0$
			\label{item:bgmlvu:lem:num_test_prob2}
		\item $\Delta(v-u) \geq 0$
			\label{item:bgmlvv-u:lem:num_test_prob2}
		\item $\mu(u)<\mu(v)$
			\label{item:mubound:lem:num_test_prob2}
		\item $0\leq\chern_1^{\beta_{-}}(u)\leq\chern_1^{\beta_{-}}(v)$
			\label{item:chern1bound:lem:num_test_prob2}
		\item $\chern_2^{\beta_{-}}(u)>0$
			\label{item:radiuscond:lem:num_test_prob2}
		\end{multicols}
	\end{enumerate}
\end{corollary}

\begin{proof}
	This is a specialization of the previous lemma, using $P=(\beta_{-},0)$.

\end{proof}


\section{B.Schmidt's Solutions to the Problems}

\subsection{Bound on \texorpdfstring{$\chern_0(u)$}{ch0(u)} for Semistabilizers}
\label{subsect:loose-bound-on-r}

The proof for the following theorem \ref{thm:loose-bound-on-r} was hinted at in
\cite{SchmidtBenjamin2020Bsot}, but the value appears explicitly in
\cite{SchmidtGithub2020}. The latter reference is a SageMath \cite{sagemath}
library for computing certain quantities related to Bridgeland stabilities on
Picard rank 1 varieties. It also includes functions to compute pseudo-walls and
pseudo-semistabilizers for tilt stability.


\begin{theorem}[Bound on $r$ - Benjamin Schmidt]
\label{thm:loose-bound-on-r}
Given a Chern character $v$ such that $\beta_-\coloneqq\beta_{-}(v)\in\QQ$, the rank $r$ of
any semistabilizer $E$ of some $F \in \firsttilt{\beta_-}$ with $\chern(F)=v$ is
bounded above by:

\begin{equation*}
	r \leq \frac{mn^2 \chern^{\beta_-}_1(v)^2}{\gcd(m,2n^2)}
\end{equation*}
\end{theorem}

\begin{proof}
The Bogomolov form applied to the twisted Chern character is the same as the
normal one. So $0 \leq \Delta(E)$ yields:

\begin{equation}
	\label{eqn-bgmlv-on-E}
	2\chern^\beta_0(E) \chern^\beta_2(E) \leq \chern^\beta_1(E)^2
\end{equation}

\noindent
Furthermore, $E \hookrightarrow F$ in $\firsttilt{\beta_{-}}$ gives:
\begin{equation}
	\label{eqn-tilt-cat-cond}
	0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)
\end{equation}
% FUTURE maybe ref this back to some definition of first tilt

\noindent
The restrictions on $\chern^{\beta_-}_0(E)$ and $\chern^{\beta_-}_2(E)$
is best seen with the following graph:

% TODO: hyperbola restriction graph (shaded)


This is where the rationality of $\beta_{-}$ comes in. If
$\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$.
Then $\chern^{\beta_-}_2(E) \in \frac{1}{\lcm(m,2n^2)}\ZZ$.
In particular, since $\chern_2^{\beta_-}(E) > 0$ (by using 	$P=(\beta_-,0)$ in
lemma \ref{lem:pseudo_wall_numerical_tests}) we must also have
$\chern^{\beta_-}_2(E) \geq \frac{1}{\lcm(m,2n^2)}$, which then in turn gives a
bound for the rank of $E$:

\begin{align}
	\chern_0(E) &= \chern^{\beta_-}_0(E) \\
	&\leq \frac{\lcm(m,2n^2) \chern^{\beta_-}_1(E)^2}{2} \\
	&= \frac{mn^2 \chern^{\beta_-}_1(F)^2}{\gcd(m,2n^2)}
\end{align}

\end{proof}

\begin{sagesilent}
from examples import recurring
\end{sagesilent}

\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-first}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=1$, $\beta_-=\sage{recurring.betaminus}$,
giving $n=\sage{recurring.n}$ and
$\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.

Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
tilt semistabilizers for $v$ are bounded above by $\sage{recurring.loose_bound}$.
However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
rank that appears turns out to be 25. This will be a recurring example to
illustrate the performance of later theorems about rank bounds
\end{example}

\begin{sagesilent}
from examples import extravagant
\end{sagesilent}

\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-first}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=1$, $\beta_-=\sage{extravagant.betaminus}$,
giving $n=\sage{extravagant.n}$ and
$\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.

Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
tilt semistabilizers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
\end{example}

\subsection{Pseudo-Wall Finding Method}

The SageMath Library \cite{SchmidtGithub2020} provides a function which
calculates all solutions to problems \ref{problem:problem-statement-1}
or \ref{problem:problem-statement-2}.
Here is an outline of the algorithm involved to do this. Simplifications will be
made in the presentation to concentrate on the case we are interested in:
problem \ref{problem:problem-statement-2}, finding all pseudo-walls when $\beta_{-}\in\QQ$.
% FUTURE add reference to section explaining new alg
In section [ref], a different
algorithm will be presented making use of the later theorems in this article,
with the goal of cutting down the run time.

\subsubsection{Finding possible \texorpdfstring{$r$}{r} and
\texorpdfstring{$c$}{c}}
To do this, first calculate the upper bound $r_{max}$ on the ranks of tilt
semistabilizers, as given by theorem \ref{thm:loose-bound-on-r}.

Recalling consequence 2 of lemma \ref{lem:pseudo_wall_numerical_tests}, we can
iterate through the possible values of $\mu(u)=\frac{c}{r}$ taking a decreasing
sequence of all fractions between $\mu(v)$ and $\beta_{-}$, who's denominators
are no large than $r_{max}$ (giving a finite sequence). This can be done with
Farey sequences \cite[chapter 6]{alma994504533502466}, for which there exist
formulae to generate.

These $\mu(u)$ values determine pairs $r,c$ up to multiples, we can then take
all multiples which satisy $0<r\leq r_{max}$.

We now have a finite sequence of pairs $r,c$ for which there might be a solution
$(r,c\ell,d\ell^2)$ to our problem. In particular, any $(r,c\ell,d\ell^2)$
satisfies consequence 2 of lemma \ref{lem:pseudo_wall_numerical_tests}, and the
positive rank condition. What remains is to find the $d$ values which satisfy
the Bogomolov inequalities and consequence 3 of lemma
\ref{lem:pseudo_wall_numerical_tests}
($\chern_2^{\beta_{-}}(u)>0$).

\subsubsection{Finding \texorpdfstring{$d$}{d} for fixed \texorpdfstring{$r$}{r}
and \texorpdfstring{$c$}{c}}

$\Delta(u) \geq 0$ induces an upper bound $\frac{c^2}{2r}$ on $d$, and the
$\chern_2^{\beta_{-}}(u)>0$ condition induces a lower bound on $d$.
The values in the range can be tested individually, to check that
the rest of the conditions are satisfied.

\subsection{Limitations}

The main downside of this algorithm is that many $r$,$c$ pairs which are tested
end up not yielding any solutions for the problem.
In fact, solutions $u$ to our problem with high rank must have $\mu(u)$ close to
$\beta_{-}$:
\begin{align*}
	0 &\leq \chern_1^{\beta_{-}}(u) \leq \chern_1^{\beta_{-}}(u) \\
	0 &\leq \mu(u) - \beta_{-} \leq \frac{\chern_1^{\beta_{-}}(v)}{r}
\end{align*}
In particular, it's the $\chern_1^{\beta_{-}}(v-u) \geq 0$ conditions which
fails for $r$,$c$ pairs with large $r$ and $\frac{c}{r}$ too far from $\beta_{-}$.
This condition is only checked within the internal loop.
This, along with a conservative estimate for a bound on the $r$ values (as
illustrated in example \ref{exmpl:recurring-first}) occasionally leads to slow
computations.

Here are some benchmarks to illustrate the performance benefits of the
alternative algorithm which will later be described in this article [ref].

\begin{center}
\begin{tabular}{ |r|l|l| }
 \hline
 Choice of $v$ on $\mathbb{P}^2$
 & $(3, 2\ell, -2)$
 & $(3, 2\ell, -\frac{15}{2})$ \\
 \hline
 \cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s & >1hr \\
 \cite{NaylorRust2023} exec time & \sim 50ms & \sim 50ms \\
 \hline
\end{tabular}
\end{center}

\section{Tighter Bounds}
\label{sec:refinement}

To get tighter bounds on the rank of destabilizers $E$ of some $F$ with some
fixed Chern character, we will need to consider each of the values which
$\chern_1^{\beta}(E)$ can take.
Doing this will allow us to eliminate possible values of $\chern_0(E)$ for which
each $\chern_1^{\beta}(E)$ leads to the failure of at least one of the inequalities.
As opposed to only eliminating possible values of $\chern_0(E)$ for which all
corresponding $\chern_1^{\beta}(E)$ fail one of the inequalities (which is what
was implicitly happening before).


First, let us fix a Chern character for $F$, and some pseudo-semistabilizer
$u$ which is a solution to problem
\ref{problem:problem-statement-1} or
\ref{problem:problem-statement-2}.
Take $\beta = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
\ref{problem:problem-statement-1}
(or $\beta = \beta_{-}$ for problem \ref{problem:problem-statement-2}).

\begin{align}
	\chern(F) =\vcentcolon\: v \:=& \:(R,C\ell,D\ell^2)
	&& \text{where $R,C\in \ZZ$ and $D\in \frac{1}{\lcm(m,2)}\ZZ$}
	\\
	u \coloneqq& \:(r,c\ell,d\ell^2)
	&& \text{where $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$}
\end{align}
 


Recall from condition \ref{item:chern1bound:lem:num_test_prob1} in
lemma \ref{lem:num_test_prob1}
(or corollary \ref{cor:num_test_prob2})
that $\chern_1^{\beta}(u)$ has fixed bounds in terms of $\chern_1^{\beta}(v)$,
and so we can write:



\begin{sagesilent}
from plots_and_expressions import c_in_terms_of_q	
\end{sagesilent}

\begin{equation}
	\label{eqn-cintermsofm}
	c=\chern_1(u) = \sage{c_in_terms_of_q}
	\qquad 0 \leq q \coloneqq \chern_1^{\beta}(u) \leq \chern_1^{\beta}(v)
\end{equation}

Furthermore, $\chern_1 \in \ZZ$ so we only need to consider
$q \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta}(F)]$,
where $n$ is the denominator of $\beta$.
For the next subsections, we consider $q$ to be fixed with one of these values,
and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.


\subsection{Numerical Inequalities}

This section studies the numerical conditions that $u$ must satisfy as per
lemma \ref{lem:num_test_prob1}
(or corollary \ref{cor:num_test_prob2})

\subsubsection{Size of pseudo-wall\texorpdfstring{: $\chern_2^P(u)>0$}{}}
\label{subsect-d-bound-radiuscond}

This condition refers to condition
\ref{item:radiuscond:lem:num_test_prob1}
from lemma \ref{lem:num_test_prob1}
(or corollary \ref{cor:num_test_prob2}).

In the case where we are tackling problem \ref{problem:problem-statement-2}
(with $\beta = \beta_{-}$), this condition, when expressed as a bound on $d$,
amounts to:

\begin{align}
\label{eqn:radius-cond-betamin}
	\chern_2^{\beta_{-}}(u) &> 0 \\
	d &> \beta_{-}q + \frac{1}{2} \beta_{-}^2r
\end{align}

\begin{sagesilent}
import other_P_choice as problem1
\end{sagesilent}

In the case where we are tackling problem \ref{problem:problem-statement-1},
with some Chern character $v$ with positive rank, and some choice of point
$P=(A,B) \in \Theta_v^-$.
Then $\sage{problem1.A2_subs}$ follows from $\chern_2^P(v)=0$. Using this substitution into the
condition $\chern_2^P(u)>0$ yields:

\begin{equation}
	\sage{problem1.radius_condition}
\end{equation}

\noindent
Expressing this as a bound on $d$, then yields:

\begin{equation}
	\sage{problem1.radius_condition_d_bound}
\end{equation}


\subsubsection{Semistability of the Semistabilizer:
	\texorpdfstring{
		$\Delta(u) \geq 0$
	}{
		Δ(u) ≥ 0
	}
}
This condition refers to condition
\ref{item:bgmlvu:lem:num_test_prob1}
from lemma \ref{lem:num_test_prob1}
(or corollary \ref{cor:num_test_prob2}).


\noindent
Expressing $\Delta(u)\geq 0$ in terms of $q$ as defined in eqn \ref{eqn-cintermsofm}
we get the following: