main.tex

%% Write  basic article template

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{color}
\usepackage{sagetex}
\usepackage{minted}
\usepackage{subcaption}
\usepackage[]{breqn}

\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\NN}{\mathbb{N}}
\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]

\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
\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}

[ref] shows that for any rational $\beta_0$,
the vertical line $\{\sigma_{\alpha,\beta_0} \colon \alpha \in \RR_{>0}\}$ only
intersects finitely many walls. 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, [ref] 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 [ref] and done
explicitly by Benjamin Schmidt within a computer program which computes
pseudowalls. 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 $v$, and a given stability condition $\sigma$, 
	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 $v$: $\mu_{\sigma}(u) = \mu_{\sigma}(v)$.

	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.

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":"x-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"$ch_2^{\alpha, \beta}(v)=0$",[3.5, 2], rotation=45, **text_args)
      + text(r"$ch_1^{\alpha, \beta}(v)=0$", [0.45, 1.5], rotation=90, **text_args)
      + text(r"$ch_2^{\alpha, \beta}(v)=0$", [-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 $\Theta_u$.
%TODO ref forwards

\subsection{Bertram's nested wall theorem}

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]
Let $v$ and $u$ be Chern characters with positive ranks and $\Delta(v),
\Delta(u)\geq 0$. Let $P$ be a point on the left branch of the characteristic
hyperbola ($\chern_2^{\alpha,\beta}(v)=0$) for $v$.

\noindent
Suppose that the following are satisfied:
\begin{itemize}
\item $u$ gives rise to a pseudo-wall for $v$, left of the characteristic
	vertical line $\chern_1^{\alpha,\beta}(v)=0$
\item The pseudo-wall contains $p$ in it's interior
	($P$ can be chosen to be the base of the left branch to target all left-walls)
\item $u$ destabilizes $v$ going `inwards', that is,
	$\nu_{\alpha,\beta}(\pm u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
	$\nu_{\alpha,\beta}(\pm u)>\nu_{\alpha,\beta}(v)$ inside.
	Where we use $+u$ or $-u$ depending on whether $(\beta,\alpha)$ is on the left
	or right (resp.) of the characteristic vertical line for $u$.
\end{itemize}

\noindent
Then we have the following:
\begin{enumerate}
	\item The pseudo-wall is left of $u$'s vertical characteristic line
		(if this is a real wall then $v$ is being semistabilized by an object with
		Chern character $u$, not $-u$)
	\item $\beta(P)<\mu(u)<\mu(v)$, i.e., $u$'s vertical characteristic line is
		positioned between $P$ and $v$'s vertical characteristic line
	\item $\chern_2^{P}(u)>0$
\end{enumerate}
Furthermore, only the last two of these consequences are sufficient to recover
all of the suppositions above.
\end{lemma}

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


For the forwards implication, assume that the suppositions of the lemma are
satisfied. The pseudo-wall intersects the characteristic hyperbola for $v$, at
some point $Q$ further up the hyperbola branch than $P$ (to satisfy second
supposition). At $Q$, we have $\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too.
This means that the characteristic hyperbola for $u$ must intersect that of $v$
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
$u$'s hyperbola, as well as the positions of the base.
However, considering the third supposition, only case 3 (green in figure) is
valid.
This is because we need $\nu_{\alpha,\beta}(u)>0$
(or $\nu_{\alpha,\beta}(-u)>0$ in case 1 involving the right hyperbola branch)
for points $(\beta,\alpha)$ on $v$'s characteristic curve inside the pseudo-wall.
In passing, note that this implies consequence 3.
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"$ch_2^{\alpha,\beta}(v)=0$")
    + line([(2,0),(2,0)] , rgbcolor = "red", legend_label=r"case 1")
    + line([(2,0),(2,0)] , rgbcolor = "blue", legend_label=r"case 2")
    + line([(2,0),(2,0)] , rgbcolor = "green", legend_label=r"case 3")
  )
  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"$ch_2^{\alpha,\beta}(v)=0$")
    + line([(2,0),(2,0)] , rgbcolor = "green",
        legend_label=r"$ch_2^{\alpha,\beta}(u)=0$")
    # vertical characteristic lines
    + line([(0,0),(0,coords_range[1][2])],
        rgbcolor="black", linestyle="dashed",
        legend_label=r"$ch_1^{\alpha,\beta}(v)=0$")
    + line([(-delta2,0),(-delta2,coords_range[1][2])],
        rgbcolor="green", linestyle="dashed",
        legend_label=r"$ch_1^{\alpha,\beta}(u)=0$")
    + line([(0,0),(-coords_range[1][2],coords_range[1][2])],
        rgbcolor="black", linestyle="dotted",
        legend_label=r"assymptote for $ch_2^{\alpha,\beta}(v)=0$")
    + line([(-delta2,0),(-delta2-coords_range[1][2],coords_range[1][2])],
        rgbcolor="green", linestyle="dotted",
        legend_label=r"assymptote for $ch_1^{\alpha,\beta}(u)=0$")
  )
  p.set_legend_options(loc="upper right")
  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 valid case (2), illustrated in Fig
\ref{fig:correct-hyperbol-intersection}.
We must have the left branch of the characteristic hyperbola for $u$ taking a
base-point to the right of that of $v$'s, but then, further up, crossing over to
the left side. The latter fact implies that the assymptote for $u$ must be to
the left of the one for $v$. Given that they are parallel and intersect the
$\beta$-axis at $\beta=\mu(u)$ and $\beta=\mu(v)$ respectively.
We must have $\mu(u)<\mu(v)$, that is, the vertical characteristic line for $u$
is to the left of the one for $v$ (consequence 2).
Finally, the fact that it is the left branch of the hyperbola for $u$ implies
consequence 1 and $\beta{P}<\mu(u)$.


Conversely, suppose that the consequences 2 and 3 are satisfied.
Consequence 2 implies that the assymptote for the left branch of the
characteristic hyperbola for $u$ is to the left of the one for $v$.
Consequence 3, along with $\beta{P}<\mu(u)$, implies that $P$ must be in the
region left of the left branch of the characteristic hyperbola for $u$.
These two facts imply that the left branch of $u$'s hyperbola is to the right of
that of $v$'s at $\alpha=\alpha(P)$, but crosses to the left side as
$\alpha \to +\infty$. This implies suppositions 1 and 2, and that the
characteristic curves for $u$ and $v$ must be in the configuration illustrated
in Fig \ref{fig:correct-hyperbol-intersection}.
Recalling consequence 3 finally confirms supposition 3.
\end{proof}

\begin{sagesilent}
v = Chern_Char(3, 2, -2)
u = Chern_Char(1, 0, 0)

def charact_curve_with_wall_plot(u,v):
    alpha = stability.Tilt().alpha
    beta = stability.Tilt().beta
    
    coords_range = (beta, -5, 5), (alpha, 0, 5)
    
    charact_curve_plot = (
      implicit_plot(stability.Tilt().degree(u), *coords_range , rgbcolor = "red")
      + implicit_plot(stability.Tilt().degree(v), *coords_range )
      + line([(mu(v),0),(mu(v),5)], linestyle = "dashed", legend_label =
      r"$(3,2\ell,-4\ell^2/2)$")
      + line([(mu(u),0),(mu(u),5)], rgbcolor = "red", linestyle =
      "dashed", legend_label = r"$(1,0,0)$")
      + implicit_plot(stability.Tilt().wall_eqn(u,v)/alpha,
      *coords_range , rgbcolor = "black")
    )
    charact_curve_plot.xmax(1)
    charact_curve_plot.xmin(-2)
    charact_curve_plot.ymax(1.5)
    charact_curve_plot.axes_labels([r"$\beta$", r"$\alpha$"])
    return charact_curve_plot
\end{sagesilent}

\begin{figure}
  \centering
	\sageplot[width=\linewidth]{charact_curve_with_wall_plot(u,v)}
	\caption{}
  \label{fig:characteristic-curves-example}
\end{figure}

Talk about figure \ref{fig:characteristic-curves-example}.


\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*}

% 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]
Given a Chern character $v$ such that $\beta_{-}(v)\in\QQ$, 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 $\beta_{-} = \frac{*}{n}$
for some $*,n \in \ZZ$.
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 $\chern_2(E) > 0$ 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} \\
	&\leq \frac{mn^2 \chern^\beta_1(F)^2}{\gcd(m,2n^2)}
\end{align}

\end{proof}

\section{B.Schmidt's Method}

Goals:
\begin{itemize}
	\item intro
	\item link repo
\end{itemize}

\subsection{Strategy}

Goals:
\begin{itemize}
	\item link repo
	\item Calc max destab rank
	\item Decrease mu(E) starting from mu(F) taking on all poss frac vals
	\item iterate over something else
	\item Stop when conditions fail
	\item method works same way for both rational beta_{-} but also for walls
		larger than certain amount
\end{itemize}

\subsection{Limitations}

Goals:
\begin{itemize}
	\item large rank forces mu to beta_{-}, so many vals of mu(E) checked
		needlessly
	\item noticeably slow (show benchmark)
\end{itemize}

\section{Refinement}
\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 $F$,
$\chern(F) = (R,C,D)$, and consider the possible Chern characters
$\chern(E) = (r,c,d)$ of some semistabilizer $E$.
 
\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 [ref] that $\chern_1^{\beta}$ has fixed bounds in terms of
$\chern(F)$, 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(E) = \sage{c_in_terms_of_q}
	\qquad 0 \leq q \leq \chern_1^{\beta}(F)
\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{
	\texorpdfstring{
		$\Delta(E) + \Delta(G) \leq \Delta(F)$
	}{
		Δ(E) + Δ(G) ≤ Δ(F)
	}
}
\label{subsect-d-bound-bgmlv1}

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:
bgmlv1 = Δ(v) - Δ(u) - Δ(v-u)
\end{sagesilent}

\begin{equation}
	\sage{0 <= bgmlv1.expand() }
\end{equation}


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

\begin{sagesilent}
bgmlv1_with_q = (
	bgmlv1
	.expand()
	.subs(c == c_in_terms_of_q)
)
\end{sagesilent}

\begin{equation}
	\sage{0 <= bgmlv1_with_q}
\end{equation}


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

\begin{sagesilent}
var("r_alt",domain="real") # r_alt = r - R/2 temporary substitution

bgmlv1_with_q_reparam = (bgmlv1_with_q.subs(r == r_alt + R/2)/r_alt).expand()

bgmlv1_d_ineq = (
	((0 >= -bgmlv1_with_q_reparam)/4 + d) # Rearrange for d
	.subs(r_alt == r - R/2) # Resubstitute r back in
	.expand()
)

bgmlv1_d_lowerbound = bgmlv1_d_ineq.rhs() # Keep hold of lower bound for d
\end{sagesilent}

\begin{dmath}
	\label{eqn-bgmlv1_d_lowerbound}
	\sage{bgmlv1_d_ineq}
\end{dmath}

\begin{sagesilent}
# Separate out the terms of the lower bound for d
bgmlv1_d_lowerbound_without_hyp = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0)

bgmlv1_d_lowerbound_exp_term = (
	bgmlv1_d_lowerbound
	- bgmlv1_d_lowerbound_without_hyp
).expand()

bgmlv1_d_lowerbound_const_term = bgmlv1_d_lowerbound_without_hyp.subs(r==0)

bgmlv1_d_lowerbound_linear_term = (
	bgmlv1_d_lowerbound_without_hyp
	- bgmlv1_d_lowerbound_const_term
).expand()

# Verify the simplified forms of the terms that will be mentioned in text
var("chbv",domain="real") # symbol to represent ch_1^\beta(v)

assert bgmlv1_d_lowerbound_const_term == (
	(
		# Keep hold of this alternative expression:
		bgmlv1_d_lowerbound_const_term_alt :=
		(
			chbv/2
			+ beta*q
		)
	)
	.subs(chbv == v.twist(beta).ch[2])
	.expand()
)

assert bgmlv1_d_lowerbound_exp_term == (
	(
		# Keep hold of this alternative expression:
		bgmlv1_d_lowerbound_exp_term_alt :=
		(
			- R*chbv/2
			- R*beta*q
			+ C*q
			- q^2
		)/(R-2*r)
	)
	.subs(chbv == v.twist(beta).ch[2])
	.expand()
)
\end{sagesilent}


\noindent
Viewing equation \ref{eqn-bgmlv1_d_lowerbound} as a lower bound for $d$ given
as a function of $r$, the terms can be rewritten as follows.
The constant term in $r$ is
$\chern^{\beta}_2(F)/2 + \beta q$.
The linear term in $r$ is
$\sage{bgmlv1_d_lowerbound_linear_term}$.
Finally, there is an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
and can be written:
$\frac{R\chern^{\beta}_2(F)/2 + R\beta q - Cq + q^2  }{2r-R}$.
In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
$\chern^{\beta}_2(F) = 0$,
so some of these expressions simplify.

\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}


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

\begin{sagesilent}
bgmlv3_with_q = (
	bgmlv3
	.expand()
	.subs(c == c_in_terms_of_q)
)
\end{sagesilent}

\begin{equation}
	\sage{0 <= bgmlv3_with_q}
\end{equation}


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

\begin{sagesilent}
var("r_alt",domain="real") # r_alt = r - R temporary substitution

bgmlv3_with_q_reparam = (
	bgmlv3_with_q
	.subs(r == r_alt + R)
	/r_alt # This operation assumes r_alt > 0
).expand()

bgmlv3_d_ineq = (
	((0 <= bgmlv3_with_q_reparam)/2 + d) # Rearrange for d
	.subs(r_alt == r - R) # Resubstitute r back in
	.expand()
)

# Check that this equation represents a bound for d
assert bgmlv3_d_ineq.lhs() == d

bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d
\end{sagesilent}

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

bgmlv3_d_upperbound_without_hyp = (
	bgmlv3_d_upperbound
	.subs(1/(R-r) == 0)
)

bgmlv3_d_upperbound_const_term = (
	bgmlv3_d_upperbound_without_hyp
	.subs(r==0)
)

bgmlv3_d_upperbound_linear_term = (
	bgmlv3_d_upperbound_without_hyp
	- bgmlv3_d_upperbound_const_term
).expand()

bgmlv3_d_upperbound_exp_term = (
	bgmlv3_d_upperbound
	- bgmlv3_d_upperbound_without_hyp
).expand()

# Verify the simplified forms of the terms that will be mentioned in text

var("chb1v chb2v",domain="real") # symbol to represent ch_1^\beta(v)
var("psi phi", domain="real") # symbol to represent ch_1^\beta(v) and
# ch_2^\beta(v)

assert bgmlv3_d_upperbound_const_term == ( 
	(
		# keep hold of this alternative expression:
		bgmlv3_d_upperbound_const_term_alt := (
			phi
			+ beta*q
		)
	)
	.subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\beta(v)
	.expand()
)

assert bgmlv3_d_upperbound_exp_term == (
	(
		# Keep hold of this alternative expression:
		bgmlv3_d_upperbound_exp_term_alt :=
		(
			R*phi
			+ (C - q)^2/2
			+ R*beta*q
			- D*R
		)/(r-R)
	)
	.subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\beta(v)
	.expand()
)

assert bgmlv3_d_upperbound_exp_term == (
	(
		# Keep hold of this alternative expression:
		bgmlv3_d_upperbound_exp_term_alt2 :=
		(
			(psi - q)^2/2/(r-R)
		)
	)
	.subs(psi == v.twist(beta).ch[1]) # subs real val of ch_1^\beta(v)
	.expand()
)
\end{sagesilent}

\bgroup
\def\psi{\chern_1^{\beta}(F)}
\def\phi{\chern_2^{\beta}(F)}
\begin{dmath}
	\label{eqn-bgmlv3_d_upperbound}
	d \leq
	\sage{bgmlv3_d_upperbound_linear_term}
	+ \sage{bgmlv3_d_upperbound_const_term_alt}
	+ \sage{bgmlv3_d_upperbound_exp_term_alt2}
\end{dmath}
\egroup


\noindent
Viewing equation \ref{eqn-bgmlv3_d_upperbound} as an upper bound for $d$ give:
as a function of $r$, the terms can be rewritten as follows.
The constant term in $r$ is
$\chern^{\beta}_2(F) + \beta q$.
The linear term in $r$ is