main.tex

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\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{color}
\usepackage{sagetex}
\usepackage{minted}
\usepackage{subcaption}
\usepackage{cancel}
\usepackage{mathtools}
\usepackage[]{breqn}
\usepackage[
backend=biber,
style=alphabetic,
sorting=ynt
]{biblatex}
\addbibresource{references.bib}

\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\chern}{\operatorname{ch}}
\newcommand{\lcm}{\operatorname{lcm}}
\newcommand{\firsttilt}[1]{\mathcal{B}^{#1}}
\newcommand{\bddderived}{\mathcal{D}^{b}}
\newcommand{\centralcharge}{\mathcal{Z}}
\newcommand{\minorheading}[1]{{\noindent\normalfont\normalsize\bfseries #1}}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corrolary}{Corrolary}[section]
\newtheorem{lemmadfn}{Lemma/Definition}[section]
\newtheorem{dfn}{Definition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{fact}{Fact}[section]
\newtheorem{example}{Example}[section]
\newtheorem{problem}{Problem Statement}

\begin{document}

\begin{sagesilent}
# Requires extra package:
#! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple

from pseudowalls import *

Δ = lambda v: v.Q_tilt()
mu = stability.Mumford().slope

def beta_minus(v):
  beta = stability.Tilt().beta
  solutions = solve(
    stability.Tilt(alpha=0).degree(v)==0,
    beta)
  return min(map(lambda s: s.rhs(), solutions))

class Object(object):
  pass
\end{sagesilent}

\title{Explicit Formulae for Bounds on the Ranks of Tilt Destabilizers and
Practical Methods for Finding Pseudowalls}

\author{Luke Naylor}

\maketitle

\tableofcontents

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

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 destabilizers
which satisfy certain numerical conditions which would be satisfied by any real
destabilizer, regardless of whether they are realised by actual semistabilizers
in $\bddderived(X)$.

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 destabilizing sequence
$E \hookrightarrow F \twoheadrightarrow G$ in $\mathcal{B}^\beta$
we have the following conditions.
There are some Bogomolov-Gieseker type inequalities:
$0 \leq \Delta(E), \Delta(G)$ and $\Delta(E) + \Delta(G) \leq \Delta(F)$.
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{Characteristic Curves of Stability Conditions Associated to Chern
Characters}

\begin{dfn}[Pseudo-semistabilizers]
	Given a Chern Character $v$ on a Picard rank 1 surface, and a given stability
	condition $\sigma_{\alpha,\beta}$,
	a pseudo-semistabilizing $u$ is a `potential' Chern character:
	\[
		u = \left(r, c\ell, d \frac{1}{2} \ell^2\right)
	\]
	which has the same tilt slope as $v$: $\nu_{\alpha,\beta}(u) = \nu_{\alpha,\beta}(v)$.

	\noindent
	Furthermore the following Bogomolov-Gieseker inequalities are satisfied:
	\begin{itemize}
		\item $\Delta(u) \geq 0$
		\item $\Delta(v-u) \geq 0$
		\item $\Delta(u) + \Delta(v-u) \leq \Delta(v)$
	\end{itemize}
	\noindent
	And finally these two conditions are satisfied:
	\begin{itemize}
		\item $\chern_1^{\beta}(u) \geq 0$
		\item $\chern_1^{\beta}(v-u) \geq 0$
	\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{dfn}

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.

\begin{lemma}[ Sanity check for Pseudo-semistabilizers ]
	Given a Picard rank 1 surface, and a given 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}

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

\begin{dfn}[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{dfn}

\begin{fact}[Geometry of Characteristic Curves]
The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
as well as the restrictions on $v$:
\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}


\minorheading{Relevance of the vertical line $V_v$}

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 $V_v$.

\begin{sagesilent}
def charact_curves(v):
    alpha = stability.Tilt().alpha
    beta = stability.Tilt().beta
    coords_range = (beta, -4, 5), (alpha, 0, 4)
    text_args = {"fontsize":"xx-large", "clip":True}
    black_text_args = {"rgbcolor": "black", **text_args}
    p = (
      implicit_plot(stability.Tilt().degree(v), *coords_range )
      + line([(mu(v),0),(mu(v),5)], linestyle = "dashed")
      + text(r"$\Theta_v^+$",[3.5, 2], rotation=45, **text_args)
      + text(r"$V_v$", [0.43, 1.5], rotation=90, **text_args)
      + text(r"$\Theta_v^-$", [-2.2, 2], rotation=-45, **text_args)
      + text(r"$\nu_{\alpha, \beta}(v)>0$", [-3, 1], **black_text_args)
      + text(r"$\nu_{\alpha, \beta}(v)<0$", [-1, 3], **black_text_args)
      + text(r"$\nu_{\alpha, \beta}(-v)>0$", [2, 3], **black_text_args)
      + text(r"$\nu_{\alpha, \beta}(-v)<0$", [4, 1], **black_text_args)
    )
    p.xmax(5)
    p.xmin(-4)
    p.ymax(4)
    p.axes_labels([r"$\beta$", r"$\alpha$"])
    return p

v1 = Chern_Char(3, 2, -2)
v2 = Chern_Char(3, 2, 2/3)
\end{sagesilent}

\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
	\centering
	\sageplot[width=\textwidth]{charact_curves(v1)}
	\caption{$\Delta(v)>0$}
	\label{fig:charact_curves_vis_bgmvlPos}
\end{subfigure}%
\hfill
\begin{subfigure}{.49\textwidth}
	\centering
	\sageplot[width=\textwidth]{charact_curves(v2)}
	\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}


\minorheading{Relevance of the hyperbola $\Theta_v$}

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 $u$, $u$ is a pseudo-semistabilizer of $v$ iff
$\nu_{\alpha,\beta}(u)=0$, 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}

\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 positive rank. Let $P$ be a point on $\Theta_v^-$.

\noindent
The following conditions:
\bgroup
\renewcommand{\labelenumi}{\alph{enumi}.}
\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}
\egroup

\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 $\chern_1^{\beta(P)}(v-u)\geq0$, $\Delta(v-u) \geq 0$
	\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}
def hyperbola_intersection_plot():
  var("alpha beta", domain="real")
  coords_range = (beta, -3, -1/2), (alpha, 0, 2.5)
  delta1 = -sqrt(2)+1/100
  delta2 = 1/2
  pbeta=-1.5
  text_args = {"fontsize":"large", "clip":True}
  black_text_args = {"rgbcolor":"black", **text_args}
  p = (
    implicit_plot( beta^2 - alpha^2 == 2,
        *coords_range , rgbcolor = "black", legend_label=r"a")
    + implicit_plot( (beta+4)^2 - (alpha)^2 == 2,
        *coords_range , rgbcolor = "red")
    + implicit_plot( (beta+delta1)^2 - alpha^2 == (delta1-2)^2-2,
        *coords_range , rgbcolor = "blue")
    + implicit_plot( (beta+delta2)^2 - alpha^2 == (delta2-2)^2-2,
        *coords_range , rgbcolor = "green")
    + point([-2, sqrt(2)], size=50, rgbcolor="black", zorder=50)
    + text("Q",[-2, sqrt(2)+0.1], **black_text_args)
    + point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor="black", zorder=50)
    + text("P",[pbeta+0.1, sqrt(pbeta^2-2)], **black_text_args)
    + circle((-2,0),sqrt(2), linestyle="dashed", rgbcolor="purple")
    # dummy lines to add legends (circumvent bug in implicit_plot)
    + line([(2,0),(2,0)] , rgbcolor = "purple", linestyle="dotted",
        legend_label=r"pseudo-wall")
    + line([(2,0),(2,0)] , rgbcolor = "black",
        legend_label=r"$\Theta_v^-$")
    + line([(2,0),(2,0)] , rgbcolor = "red", legend_label=r"$\Theta_u$ case 1")
    + line([(2,0),(2,0)] , rgbcolor = "blue", legend_label=r"$\Theta_u$ case 2")
    + line([(2,0),(2,0)] , rgbcolor = "green", legend_label=r"$\Theta_u$ case 3")
  )
  p.set_legend_options(loc="upper right", font_size="x-large",
    font_family="serif")
  p.xmax(coords_range[0][2])
  p.xmin(coords_range[0][1])
  p.ymax(coords_range[1][2])
  p.ymin(coords_range[1][1])
  p.axes_labels([r"$\beta$", r"$\alpha$"])
  return p

def correct_hyperbola_intersection_plot():
  var("alpha beta", domain="real")
  coords_range = (beta, -2.5, 0.5), (alpha, 0, 3)
  delta2 = 1/2
  pbeta=-1.5
  text_args = {"fontsize":"large", "clip":True}
  black_text_args = {"rgbcolor":"black", **text_args}
  p = (
    implicit_plot( beta^2 - alpha^2 == 2,
        *coords_range , rgbcolor = "black", legend_label=r"a")
    + implicit_plot((beta+delta2)^2 - alpha^2 == (delta2-2)^2-2,
        *coords_range , rgbcolor = "green")
    + point([-2, sqrt(2)], size=50, rgbcolor="black", zorder=50)
    + text("Q",[-2, sqrt(2)+0.1], **black_text_args)
    + point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor="black", zorder=50)
    + text("P",[pbeta+0.1, sqrt(pbeta^2-2)], **black_text_args)
    + circle((-2,0),sqrt(2), linestyle="dashed", rgbcolor="purple")
    # dummy lines to add legends (circumvent bug in implicit_plot)
    + line([(2,0),(2,0)] , rgbcolor = "purple", linestyle="dotted",
        legend_label=r"pseudo-wall")
    + line([(2,0),(2,0)] , rgbcolor = "black",
        legend_label=r"$\Theta_v^-$")
    + line([(2,0),(2,0)] , rgbcolor = "green",
        legend_label=r"$\Theta_u^-$")
    # vertical characteristic lines
    + line([(0,0),(0,coords_range[1][2])],
        rgbcolor="black", linestyle="dashed",
        legend_label=r"$V_v$")
    + line([(-delta2,0),(-delta2,coords_range[1][2])],
        rgbcolor="green", linestyle="dashed",
        legend_label=r"$V_u$")
    + line([(-delta2,0),(-delta2-coords_range[1][2],coords_range[1][2])],
        rgbcolor="green", linestyle="dotted",
        legend_label=r"$\Theta_u^-$ assymptote")
    + line([(0,0),(-coords_range[1][2],coords_range[1][2])],
        rgbcolor="black", linestyle="dotted",
        legend_label=r"$\Theta_v^-$ assymptote")
  )
  p.set_legend_options(loc="upper right", font_size="x-large",
    font_family="serif")
  p.xmax(coords_range[0][2])
  p.xmin(coords_range[0][1])
  p.ymax(coords_range[1][2])
  p.ymin(coords_range[1][1])
  p.axes_labels([r"$\beta$", r"$\alpha$"])
  return p
\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.
\end{proof}


\section{Loose Bounds on $\chern_0(E)$ for Semistabilizers Along Fixed
$\beta\in\QQ$}

\begin{dfn}[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{dfn}

$\chern^\beta_1(E)$ is the imaginary component of the central charge
$\centralcharge_{\alpha,\beta}(E)$ and any element of $\firsttilt\beta$
satisfies $\chern^\beta_1 \geq 0$. This, along with additivity gives us, for any
destabilizing sequence [ref]:
\begin{equation}
	\label{eqn-tilt-cat-cond}
	0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)
\end{equation}

When finding Chern characters of potential destabilizers $E$ for some fixed
Chern character $\chern(F)$, this bounds $\chern_1(E)$.

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}

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

% TODO: hyperbola restriction graph (shaded)
\begin{sagesilent}
var("m") # Initialize symbol for variety parameter
\end{sagesilent}

This is where the rationality of $\beta_{-}$ comes in. If
$\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$. Then
$\chern^{\beta_-}_2(E) \in \frac{1}{\lcm(m,2n^2)}\ZZ$ where $m$ is the integer
which guarantees $\chern_2(E) \in \frac{1}{m}\ZZ$ (determined by the variety).
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{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-first}
\begin{sagesilent}
recurring = Object()
recurring.chern = Chern_Char(3, 2, -2)
recurring.b = beta_minus(recurring.chern)
recurring.twisted = recurring.chern.twist(recurring.b)
\end{sagesilent}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta_-=\sage{recurring.b}$,
giving $n=\sage{recurring.b.denominator()}$ and
$\chern_1^{\sage{recurring.b}}(F) = \sage{recurring.twisted.ch[1]}$.

\begin{sagesilent}
n = recurring.b.denominator()
m = 2
loose_bound = (
  m*n^2*recurring.twisted.ch[1]^2
) / gcd(m, 2*n^2)
\end{sagesilent}
Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
tilt semistabilizers for $v$ are bounded above by $\sage{loose_bound}$.
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{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-first}
\begin{sagesilent}
extravagant = Object()
extravagant.chern = Chern_Char(29, 13, -3/2)
extravagant.b = beta_minus(extravagant.chern)
extravagant.twisted = extravagant.chern.twist(extravagant.b)
extravagant.actual_rmax = 49313
\end{sagesilent}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta_-=\sage{extravagant.b}$,
giving $n=\sage{extravagant.b.denominator()}$ and
$\chern_1^{\sage{extravagant.b}}(F) = \sage{extravagant.twisted.ch[1]}$.

\begin{sagesilent}
n = extravagant.b.denominator()
m = 2
loose_bound = (
  m*n^2*extravagant.twisted.ch[1]^2
) / gcd(m, 2*n^2)
\end{sagesilent}
Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
tilt semistabilizers for $v$ are bounded above by $\sage{loose_bound}$.
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}

\section{B.Schmidt's Wall Finding Method}

The proof for the previous theorem 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.

Here is an outline of the algorithm involved to do this. Simplifications will be
made in the presenteation to concentrate on the case we are interested in:
finding all pseudo-walls when $\beta_{-}\in\QQ$. 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.

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

Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
and $\beta_{-}(v) \in \QQ$.
The goal is to find all pseudo-semistabilizers $u=(r,c\ell,d\ell^2)$
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
(`outward' destabilizers can be recovered as $v-u$).

This will give all pseudo-walls between the chamber corresponding to Gieseker
stability and the stability condition corresponding to $P$.
\end{problem}

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

Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
and $\beta_{-}(v) \in \QQ$.
The goal is to find all solutions $u=(r,c\ell,d\ell^2)$
to problem \ref{problem:problem-statement-1} with the choice
$P=(\beta_{-},0)$.

This will give all circular pseudo-walls left of $V_v$.
\end{problem}

\subsection*{Algorithm}

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

$\Delta(u) \geq 0$ 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 semistabilizer $E$:
\begin{align}
	v &\coloneqq \chern(F) = (R,C\ell,D\ell^2) \\
	u &\coloneqq \chern(E) = (r,c\ell,d\ell^2)
\end{align}
 
\begin{sagesilent}
# Requires extra package:
#! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple

from pseudowalls import *

v = Chern_Char(*var("R C D", domain="real"))
u = Chern_Char(*var("r c d", domain="real"))

Δ = lambda v: v.Q_tilt()
\end{sagesilent}

Recall from eqn \ref{eqn-tilt-cat-cond} that $\chern_1^{\beta}(u)$ has fixed
bounds in terms of $\chern_1^{\beta}(v)$, and so we can write:

\begin{sagesilent}
ts = stability.Tilt
var("beta", domain="real")

c_lower_bound = -(
	ts(beta=beta).rank(u)
	/ts().alpha
).expand() + c

var("q", domain="real")
c_in_terms_of_q = c_lower_bound + 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
$q \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta}(F)]$.
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}

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

This condition refers to consequence 4 from
lemma \ref{lem:pseudo_wall_numerical_tests}.

In the case where $P$ was chosen to be the base of $\Theta_v^-$: $P=(\beta_{-},0)$.
This condition amounts to:

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

\subsubsection{
	\texorpdfstring{
		$\Delta(u,v-u) \geq 0$
	}{
		Δ(u,v-u) ≤ 0
	}
}
\label{subsect-d-bound-bgmlv1}

Writing the condition in terms of the twisted chern characters
for $u$ and $v$ at $\beta$
($(r,\chern_1^{\beta}(u),\chern_2^{\beta}(u))$
and $(R-r,\chern_1^{\beta}(v-u),\chern_2^{\beta}(v-u))$) yields:

\begin{equation}
\label{eqn:bgmlv1-pt1}
	(R-r)\chern_2^{\beta}(u)
	\leq
	\chern_1^{\beta}(u)\chern_1^{\beta}(v-u)
	- r\chern_2^{\beta}(v-u)
\end{equation}

Which rearranges to (using additivity of $\chern_2^{\beta}$):

\begin{equation}
\label{eqn:bgmlv1-pt2}
	(R-2r)\chern_2^{\beta}(u)
	\leq
	\chern_1^{\beta}(u)\chern_1^{\beta}(v-u)
	- r\chern_2^{\beta}(v)
\end{equation}

With $u$ satisfying the condition given by equation \ref{eqn-cintermsofm},
we note that $\chern_1^{\beta}(u),\chern_1^{\beta}(v-u) \geq 0$.

In the special case with $P=(\beta_{-},0)$,
we have $\chern_2^{\beta_{-}}(v)=0$, and we can assume
equation $\chern_2^{\beta_{-}}(u)>0$ (eqn \ref{eqn:radius-cond-betamin})
in the context of our problem.

Finally, $r>0$ as per the statement of the problem, so the right-hand-side
of equation \ref{eqn:bgmlv1-pt1} is always greater than, or equal, to zero.
And so, when $P\coloneqq(\beta_{-},0)$, this condition $\Delta(u,v-u) \geq 0$ is
always satisfied when $2r \geq R$, provided that the other conditions of the
problem statement (\ref{subsect:problem-statement}) hold.

However, when $2r<R$, this condition does add potentially independent condition
of the others:

\begin{equation}
\label{eqn:bgmlv1-pt3}
	\chern_2^{\beta}(u)
	\leq
	\frac{
		\chern_1^{\beta}(u)\chern_1^{\beta}(v-u)
		- r\chern_2^{\beta}(v)
	}
	{R-2r},
	\qquad
	2r<R
\end{equation}

Expressed in terms of $d$ and $q$:
\begin{equation}
\label{eqn:bgmlv1-pt4}
	d
	\leq
	\beta_{-}q
	+\frac{1}{2}{\beta_{-}}^2r
	+
	\frac{
		q(\chern_1^{\beta}(v)-q)
		- r\chern_2^{\beta}(v)
	}
	{R-2r},
	\qquad
	2r<R
\end{equation}

\subsubsection{
	\texorpdfstring{
		$\Delta(E) \geq 0$
	}{
		Δ(E) ≥ 0
	}
}

This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:

\begin{sagesilent}
# First Bogomolov-Gieseker form expression that must be non-negative:
bgmlv2 = Δ(u)
\end{sagesilent}

\begin{equation}
	\sage{0 <= bgmlv2.expand() }
\end{equation}


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

\begin{sagesilent}
bgmlv2_with_q = (
	bgmlv2
	.expand()
	.subs(c == c_in_terms_of_q)
)
\end{sagesilent}

\begin{equation}
	\sage{0 <= bgmlv2_with_q}
\end{equation}


\noindent
This can be rearranged to express a bound on $d$ as follows:

\begin{sagesilent}
bgmlv2_d_ineq = (
	(0 <= bgmlv2_with_q)/2/r # rescale assuming r > 0
	+ d # Rearrange for d
).expand()

# Keep hold of lower bound for d
bgmlv2_d_upperbound = bgmlv2_d_ineq.rhs()
\end{sagesilent}

\begin{equation}
	\label{eqn-bgmlv2_d_upperbound}
	\sage{bgmlv2_d_ineq}
\end{equation}

\begin{sagesilent}
# Seperate out the terms of the lower bound for d

bgmlv2_d_upperbound_without_hyp = (
	bgmlv2_d_upperbound
	.subs(1/r == 0)
)

bgmlv2_d_upperbound_const_term = (
	bgmlv2_d_upperbound_without_hyp
	.subs(r==0)
)

bgmlv2_d_upperbound_linear_term = (
	bgmlv2_d_upperbound_without_hyp
	- bgmlv2_d_upperbound_const_term
).expand()

bgmlv2_d_upperbound_exp_term = (
	bgmlv2_d_upperbound
	- bgmlv2_d_upperbound_without_hyp
).expand()
\end{sagesilent}

Viewing equation \ref{eqn-bgmlv2_d_upperbound} as a lower bound for $d$ in term
of $r$ again, there is a constant term
$\sage{bgmlv2_d_upperbound_const_term}$,
a linear term
$\sage{bgmlv2_d_upperbound_linear_term}$,
and a hyperbolic term
$\sage{bgmlv2_d_upperbound_exp_term}$.
Notice that for $\beta = \beta_{-}$ (or $\beta_{+}$), that is when
$\chern^{\beta}_2(F)=0$, the constant and linear terms match up with the ones
for the bound found for $d$ in subsection \ref{subsect-d-bound-bgmlv1}.

\subsubsection{
	\texorpdfstring{
		$\Delta(G) \geq 0$
	}{
		Δ(G) ≥ 0
	}
}
\label{subsect-d-bound-bgmlv3}

This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:

\begin{sagesilent}
# Third Bogomolov-Gieseker form expression that must be non-negative:
bgmlv3 = Δ(v-u)
\end{sagesilent}

\begin{equation}
	\sage{0 <= bgmlv3.expand() }
\end{equation}