\section{Existing Bound on Semistabiliser Ranks}
\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} as shown in the following Listing
\ref{fig:code:schmidt-bound}.
The latter citation 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-semistabilisers for tilt stability.

\lstinputlisting[
	caption={\raggedleft snippet from \cite{SchmidtGithub2020} \texttt{stability_conditions.tilt.walls_left}},
	label={fig:code:schmidt-bound},
	linerange=1-7,
	firstnumber=1015
]{schmidt-snippet}

\begin{theorem}[Bound on $r$ - Benjamin Schmidt]
\label{thm:loose-bound-on-r}
Let $X$ be a smooth projective Picard rank 1 surface with choice of ample line
bundle $L$ such that $\ell\coloneqq c_1(L)$ generates $\neronseveri(X)$ and
take $m\coloneqq \ell^2$.

Given a Chern character $v$ with $\Delta(v) \geq 0$ and positive rank (or
$\chern_0(v) = 0$ but $\chern_1(v) > 0$)
such that
$\beta_-\coloneqq\beta_{-}(v)\in\QQ$, the rank $r$ of
any solution $u$ of Problem \ref{problem:problem-statement-2} is
bounded above by:

\begin{equation*}
	r \leq \frac{m\left(n \chern^{\beta_-}_1(v) - 1\right)^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
untwisted one. So $0 \leq \Delta(u)$
(condition 2 from Corollary \ref{cor:num_test_prob2})
yields:

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

\noindent
Furthermore,
condition 5 from Corollary \ref{cor:num_test_prob2}
gives:
\begin{equation}
	\label{eqn-tilt-cat-cond}
	0 < \chern^\beta_1(u) < \chern^\beta_1(v)
\end{equation}
% FUTURE maybe ref this back to some definition of first tilt

\noindent
The restrictions on $\chern^{\beta_-}_0(u)$ and $\chern^{\beta_-}_2(v)$
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) \nonumber \\
	&\leq \frac{\lcm(m,2n^2) \chern^{\beta_-}_1(E)^2}{2} \nonumber \\
	&= \frac{mn^2 \chern^{\beta_-}_1(F)^2}{\gcd(m,2n^2)}
	\label{proof:first-bound-on-r}
\end{align}

In fact Equation \ref{eqn-tilt-cat-cond} can be tightened slightly:
we cannot have equality $\chern^{\beta_{-}}_1(E) = \chern^{\beta_{-}}_1(F)$
otherwise we would have $\chern^{\beta_{-}}_1(G)=0$ for the quotient $G$.
This would imply $\mu(G)=\beta_{-}$, but since $\Theta_G$ is bounded above in the
upper-half plane by the assymptotes crossing the $\beta$-axis at $\pm45^\circ$ at
$\beta=\beta_{-}(v)$. So $\Theta_G$ cannot intersect $\Theta_v$ at any point
with $\alpha > 0$, so there is no point with $\nu(E)=\nu(F)=\nu(G)=0$, which would
have to hold at the top of the pseudo-wall if it were to exist.
Therefore we must have a strict inequality
$\chern^{\beta_{-}}_1(E) < \chern^{\beta_{-}}_1(F)$,
and since these are elements of $\frac{1}{n}\ZZ$, we can also conclude:
\[
	n\chern^{\beta_{-}}_1(E) \leq n\chern^{\beta_{-}}_1(F) - 1
\]
which then tightens the upper bound found for $\chern_0(E)$
in Equation \ref{proof:first-bound-on-r}
to the bound in the statement of the Lemma.

\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 semistabilisers for $v$ are bounded above by $\sage{recurring.loose_bound}$.
However, when computing all tilt semistabilisers 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 semistabilisers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
However, when computing all tilt semistabilisers for $v$ on $\PP^2$, the maximum
rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
\end{example}


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

To get tighter bounds on the rank of solutions $u$ to the Problems
\ref{problem:problem-statement-1} and
\ref{problem:problem-statement-2},
we will need to consider each of the values which
$\chern_1^{\beta}(u)$ can take.
Doing this will allow us to eliminate possible values of $\chern_0(u)$ for which
each $\chern_1^{\beta}(u)$ leads to the failure of at least one of the inequalities.
As opposed to only eliminating possible values of $\chern_0(u)$ for which all
corresponding $\chern_1^{\beta}(u)$ fail one of the inequalities (which is what
was implicitly happening before).


First, let us fix a Chern character $v$ with $\Delta(v)\geq 0$,
$\chern_0(v)>0$, or $\chern_0(v)=0$ and $\chern_1(v)>0$,
and some solution $u$ to the 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 $D\in \frac{1}{\lcm(m,2)}\ZZ$}
	\nonumber
	\\
	u \coloneqq& \:(r,c\ell,d\ell^2)
	&& \text{where $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$}
	\label{eqn:u-coords}
\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, if $\beta$ is rational, $\chern_1(u) \in \ZZ$ 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{Bounds on \texorpdfstring{`$d$'}{d}-values for Solutions of Problems}

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})
and reformulates them as bounds on $d$ from Equation \ref{eqn:u-coords}.
This is done to determine which $r$ values lead to no possible values for $d$.

\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 &> \frac{1}{2} {\beta_{-}}^2r + \beta_{-}q
	\nonumber
\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:


\begin{sagesilent}
from plots_and_expressions import bgmlv2_with_q
\end{sagesilent}

\begin{equation}
	\sage{bgmlv2_with_q}
\end{equation}


\noindent
This can be rearranged to express a bound on $d$ as follows
(recall from condition \ref{item:rankpos:lem:num_test_prob1}
in Lemma \ref{lem:num_test_prob1} or corollary
\ref{cor:num_test_prob2} that $r>0$):


\begin{sagesilent}
from plots_and_expressions import bgmlv2_d_ineq
\end{sagesilent}
\begin{equation}
	\label{eqn-bgmlv2_d_upperbound}
	\sage{bgmlv2_d_ineq}
\end{equation}

\begin{sagesilent}
from plots_and_expressions import bgmlv2_d_upperbound_terms
\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
$\sage{bgmlv2_d_upperbound_terms.const}$,
a linear term
$\sage{bgmlv2_d_upperbound_terms.linear}$,
and a hyperbolic term
$\sage{bgmlv2_d_upperbound_terms.hyperbolic}$.
Notice that in the context of problem \ref{problem:problem-statement-2}
($\beta = \beta_{-}$),
the constant and linear terms match up with the ones
for the bound found for $d$ in subsubsection \ref{subsect-d-bound-radiuscond}.

\subsubsection{Semistability of the Quotient:
	\texorpdfstring{
		$\Delta(v-u) \geq 0$
	}{
		Δ(v-u) ≥ 0
	}
}
\label{subsect-d-bound-bgmlv3}

This condition refers to condition
\ref{item:bgmlvv-u:lem:num_test_prob1}
from Lemma \ref{lem:num_test_prob1}
(or corollary \ref{cor:num_test_prob2}).

Expressing $\Delta(v-u)\geq 0$ in term of $q$ and rearranging as a bound on
$d$ yields:


\begin{sagesilent}
from plots_and_expressions import bgmlv3_d_upperbound_terms
\end{sagesilent}

\begin{equation*}
	\label{eqn-bgmlv3_d_upperbound}
	d \leq
	\sage{bgmlv3_d_upperbound_terms.linear}
	+ \sage{bgmlv3_d_upperbound_terms.const}
	+ \sage{bgmlv3_d_upperbound_terms.hyperbolic}
	\qquad
	\text{where }r>R
\end{equation*}


\noindent
For $r=R$, $\Delta(v-u)\geq 0$ is always true, and for $r<R$ it gives a lower
bound on $d$, but it is weaker than the one given by the lower bound in
subsubsection \ref{subsect-d-bound-radiuscond}.
Viewing the right hand side of Equation \ref{eqn-bgmlv3_d_upperbound}
as a function of $r$, the linear and constant terms almost match up with the
ones in the previous section, up to the 
$\chern_2^{\beta}(v)$ term.


However, when specializing to problem \ref{problem:problem-statement-2} again
(with $\beta = \beta_{-}$), then we have $\chern^{\beta}_2(v) = 0$.
And so in this context, the linear and constant terms do match up with the
previous subsubsections.

\subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
\texorpdfstring{\ref{problem:problem-statement-2}}{2}}
\label{subsubsect:all-bounds-on-d-prob2}
%% RECAP ON INEQUALITIES TOGETHER

%%%% RATIONAL BETA MINUS
As mentioned in passing, when specializing to solutions $u$ of problem
\ref{problem:problem-statement-2}, the bounds on
$d=\chern_2(u)$ induced by conditions
\ref{item:bgmlvu:lem:num_test_prob2},
\ref{item:bgmlvv-u:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob1}
from corollary \ref{cor:num_test_prob2} have the same constant and linear
terms in $r$, but different hyperbolic terms.
These give bounds with the same assymptotes when we take $r\to\infty$
(for any fixed $q=\chern_1^{\beta_{-}}(u)$).

% redefine \beta (especially coming from rendered SageMath expressions)
% to be \beta_{-} for the rest of this subsubsection
\bgroup

\let\originalbeta\beta
\renewcommand\beta{{\originalbeta_{-}}}

\begin{align}
	d &>&
	\frac{1}{2}\beta^2 r
	&+ \beta q,
	\phantom{+}& % to keep terms aligned
	 &\qquad\text{when\:} r > 0
	\label{eqn:radiuscond_d_bound_betamin}
\\
	d &\leq&
	\sage{bgmlv2_d_upperbound_terms.problem2.linear}
	&+ \sage{bgmlv2_d_upperbound_terms.problem2.const}
	+& \sage{bgmlv2_d_upperbound_terms.problem2.hyperbolic},
	 &\qquad\text{when\:} r > 0
	 \label{eqn:bgmlv2_d_bound_betamin}
\\
	d &\leq&
	\sage{bgmlv3_d_upperbound_terms.problem2.linear}
	&+ \sage{bgmlv3_d_upperbound_terms.problem2.const}
	% ^ ch_2^\beta(F)=0 for beta_{-}
	+& \sage{bgmlv3_d_upperbound_terms.problem2.hyperbolic},
	 &\qquad\text{when\:} r > R
	 \label{eqn:bgmlv3_d_bound_betamin}
\end{align}


\begin{sagesilent}
from plots_and_expressions import \
bounds_on_d_qmin, \
bounds_on_d_qmax
\end{sagesilent}

\begin{figure}
\centering
\begin{subfigure}{.45\textwidth}
  \centering
	\sageplot[width=\linewidth]{bounds_on_d_qmin}
	\caption{$q = 0$ (all bounds other than green coincide on line)}
  \label{fig:d_bounds_xmpl_min_q}
\end{subfigure}%
\hfill
\begin{subfigure}{.45\textwidth}
  \centering
	\sageplot[width=\linewidth]{bounds_on_d_qmax}
	\caption{$q = \chern^{\beta}(F)$ (all bounds other than blue coincide on line)}
  \label{fig:d_bounds_xmpl_max_q}
\end{subfigure}
\caption{
	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq\chern_0(E)$ for fixed, extreme,
	values of $q\coloneqq\chern_1^{\beta}(E)$.
	Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_extrm_q}
\end{figure}

Recalling that $q \coloneqq \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
it is worth noting that the extreme values of $q$ in this range lead to the
tightest bounds on $d$, as illustrated in Figure
(\ref{fig:d_bounds_xmpl_extrm_q}).
In fact, in each case, one of the weak upper bounds coincides with one of the
weak lower bounds, (implying no possible destabilisers $E$ with
$\chern_0(E)=\vcentcolon r>R\coloneqq\chern_0(F)$ for these $q$-values).
This indeed happens in general since the right hand sides of
(eqn \ref{eqn:bgmlv2_d_bound_betamin}) and
(eqn \ref{eqn:radiuscond_d_bound_betamin}) match when $q=0$.
In the other case, $q=\chern^{\beta}_1(F)$, it is the right hand sides of
(eqn \ref{eqn:bgmlv3_d_bound_betamin}) and
(eqn \ref{eqn:radiuscond_d_bound_betamin}) which match.


The more generic case, when $0 < q\coloneqq\chern_1^{\beta}(E) < \chern_1^{\beta}(F)$
for the bounds on $d$ in terms of $r$ is illustrated in Figure
(\ref{fig:d_bounds_xmpl_gnrc_q}).
The question of whether there are pseudo-destabilisers of arbitrarily large
rank, in the context of the graph, comes down to whether there are points
$(r,d) \in \ZZ \oplus \frac{1}{\lcm(m,2)} \ZZ$ (with large $r$)
that fit above the yellow line (ensuring positive radius of wall) but below the
blue and green (ensuring $\Delta(u), \Delta(v-u) > 0$).
These lines have the same assymptote at $r \to \infty$
(eqns \ref{eqn:bgmlv2_d_bound_betamin},
\ref{eqn:bgmlv3_d_bound_betamin},
\ref{eqn:radiuscond_d_bound_betamin}).
As mentioned in the introduction to this Part, the finiteness of these
solutions is entirely determined by whether $\beta$ is rational or irrational.
Some of the details around the associated numerics are explored next.

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

\begin{figure}
\centering
\sageplot[width=\linewidth]{typical_bounds_on_d}
\caption{
	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq \chern_0(E)$ for a fixed
	value $\chern_1^{\beta}(F)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
	Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_gnrc_q}
\end{figure}

\subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
\texorpdfstring{\ref{problem:problem-statement-1}}{1}}
\label{subsubsect:all-bounds-on-d-prob1}

Unlike for problem \ref{problem:problem-statement-2},
the bounds on $d=\chern_2(u)$ induced by conditions
\ref{item:bgmlvu:lem:num_test_prob2},
\ref{item:bgmlvv-u:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob1}
from corollary \ref{cor:num_test_prob2} have different
constant and linear terms, so that the graphs for upper
bounds do not share the same assymptote as the lower bound
(and they will turn out to intersect).

\begin{align}
	\sage{problem1.radius_condition_d_bound.lhs()}
	&>
	\sage{problem1.radius_condition_d_bound.rhs()}
	&\text{where }r>0
	\label{eqn:prob1:radiuscond}
	\\
	d &\leq
	\sage{problem1.bgmlv2_d_upperbound_terms.linear}
	+ \sage{problem1.bgmlv2_d_upperbound_terms.const}
	+ \sage{problem1.bgmlv2_d_upperbound_terms.hyperbolic}
	&\text{where }r>R
	\label{eqn:prob1:bgmlv2}
	\\
	d &\leq
	\sage{problem1.bgmlv3_d_upperbound_terms.linear}
	+ \sage{problem1.bgmlv3_d_upperbound_terms.const}
	+ \sage{problem1.bgmlv3_d_upperbound_terms.hyperbolic}
	&\text{where }r>R
	\label{eqn:prob1:bgmlv3}
\end{align}

Notice that as a function in $r$, the linear term in 
equation \ref{eqn:prob1:radiuscond} is strictly greater than
those in Equations \ref{eqn:prob1:bgmlv2}
and \ref{eqn:prob1:bgmlv3}. This is because $r$, $R$
and $\chern_2^B(v)$ are all strictly positive:
\begin{itemize}
	\item $R > 0$ from the setting of problem
	\ref{problem:problem-statement-1}
	\item $r > 0$ from Lemma \ref{lem:num_test_prob1}
	\item $\chern_2^B(v)>0$ because $B < \originalbeta_{-}$ due to the choice of $P$ being
	a point on $\Theta_v^{-}$
\end{itemize}

This means that the lower bound for $d$ will be large than either of the two
upper bounds for sufficiently large $r$, and hence those values of $r$ would yield no 
solution to problem \ref{problem:problem-statement-1}.

A generic example of this is plotted in Figure
\ref{fig:problem1:d_bounds_xmpl_gnrc_q}.

\begin{figure}
\centering
\sageplot[width=\linewidth]{problem1.example_plot}
\caption{
	Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq \chern_0(E)$ for a fixed
	value $\chern_1^{\beta}(F)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
	Where $\chern(F) = (3,2,-2)$ and $P$ chosen as the point on $\Theta_v$
	with $B\coloneqq-2/3-1/99$ in the context of problem 
	\ref{problem:problem-statement-1}.
}
\label{fig:problem1:d_bounds_xmpl_gnrc_q}
\end{figure}

\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{} in Problem
\ref{problem:problem-statement-1}}

As discussed at the end of subsection \ref{subsubsect:all-bounds-on-d-prob1}
(and illustrated in Figure \ref{fig:problem1:d_bounds_xmpl_gnrc_q}),
there are no solutions $u$ to problem \ref{problem:problem-statement-1}
with large $r=\chern_0(u)$, since the lower bound on $d=\chern_2(u)$ is larger
than the upper bounds.
Therefore, we can calculate upper bounds on $r$ by calculating for which values,
the lower bound on $d$ is equal to one of the upper bounds on $d$
(i.e. finding certain intersection points of the graph in Figure
\ref{fig:problem1:d_bounds_xmpl_gnrc_q}).

\begin{theorem}[Problem \ref{problem:problem-statement-1} upper Bound on $r$]
\label{lem:prob1:r_bound}
	Let $u$ be a solution to problem \ref{problem:problem-statement-1}
	and $q\coloneqq\chern_1^{B}(u)$.
	Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression:
	\begin{equation}
		\sage{problem1.r_bound_expression}
	\end{equation}
\end{theorem}

\begin{proof}
	Recall that $d\coloneqq\chern_2(u)$ has two upper bounds in terms of $r$: in
	Equations \ref{eqn:prob1:bgmlv2} and \ref{eqn:prob1:bgmlv3};
	and one lower bound: in Equation \ref{eqn:prob1:radiuscond}.

	Solving for the lower bound in Equation \ref{eqn:prob1:radiuscond} being
	less than the upper bound in Equation \ref{eqn:prob1:bgmlv2} yields:
	\begin{equation}
	r<\sage{problem1.positive_intersection_bgmlv2}
	\end{equation}

	Similarly, but with the upper bound in Equation \ref{eqn:prob1:bgmlv3}, gives:
	\begin{equation}
	r<\sage{problem1.positive_intersection_bgmlv3}
	\end{equation}

	Therefore, $r$ is bounded above by the minimum of these two expressions which
	can then be factored into the expression given in the Lemma.
	
\end{proof}

The above Lemma \ref{lem:prob1:r_bound} gives an upper bound on $r$ in terms of $q$.
But given that $0 \leq q \leq \chern_1^{B}(v)$, we can take the maximum of this
bound, over $q$ in this range, to get a simpler (but weaker) bound in the
following Lemma \ref{lem:prob1:convenient_r_bound}.

\begin{theorem}[Problem \ref{problem:problem-statement-1} global upper Bound on $r$]
\label{lem:prob1:convenient_r_bound}
	Let $u$ be a solution to problem \ref{problem:problem-statement-1}.
	Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression:
	\begin{equation}
		\sage{problem1.r_max}
	\end{equation}
\end{theorem}

\begin{proof}
	The first term of the minimum in Lemma \ref{lem:prob1:r_bound}
	increases linearly in $q$, and the second
	decreases linearly. So the maximum is achieved with the value of
	$q=q_{\mathrm{max}}$ where they are equal.
	Solving for the two terms in the minimum to be equal yields:
	$q_{\mathrm{max}}=\sage{problem1.maximising_q}$.
	Substituting $q=q_{\mathrm{max}}$ into the bound in Lemma
	\ref{lem:prob1:r_bound} gives the bound as stated in the current Lemma.
	
\end{proof}

\begin{remark}
	$q_{\mathrm{max}} > 0$ is immediate from the expression, but
	$q_{\mathrm{max}} \leq \chern_1^{B}(v)$ is equivalent to $\Delta(v) \geq 0$,
	which is true by assumption in this setting.
\end{remark}


\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{} in Problem
\ref{problem:problem-statement-2}}

Now, the inequalities from the above subsubsection
\ref{subsubsect:all-bounds-on-d-prob2} will be used to find, for
each given $q=\chern^{\beta}_1(E)$, how large $r$ needs to be in order to leave
no possible solutions for $d$. At that point, there are no solutions
$u=(r,c\ell,d\ell^2)$ to problem \ref{problem:problem-statement-2}.
The strategy here is similar to what was shown in Theorem
\ref{thm:loose-bound-on-r}.


\renewcommand{\aa}{{a_v}}
\newcommand{\bb}{{b_q}}
Suppose $\beta = \frac{\aa}{n}$ for some coprime $n \in \NN,\aa \in \ZZ$.
Then fix a value of $q$:
\begin{equation}
	q\coloneqq \chern_1^{\beta}(E)
	  =\frac{\bb}{n}
	\in
	\frac{1}{n} \ZZ
	\cap [0, \chern_1^{\beta}(F)]
\end{equation}
as noted at the beginning of this section \ref{sec:refinement} so that we are
considering $u$ which satisfy \ref{item:chern1bound:lem:num_test_prob2}
in corollary \ref{cor:num_test_prob2}.

Substituting the current values of $q$ and $\beta$ into the condition for the
radius of the pseudo-wall being positive
(eqn \ref{eqn:radiuscond_d_bound_betamin}) we get:

\begin{sagesilent}
from plots_and_expressions import \
positive_radius_condition_with_q, \
q_value_expr, \
beta_value_expr
\end{sagesilent}
\begin{equation}
\label{eqn:positive_rad_condition_in_terms_of_q_beta}
	\frac{1}{\lcm(m,2)}\ZZ
	\ni
	\qquad
	\sage{positive_radius_condition_with_q.subs([q_value_expr,beta_value_expr]).factor()}
	\qquad
	\in
	\frac{1}{2n^2}\ZZ
\end{equation}


\begin{sagesilent}
from plots_and_expressions import main_theorem1
\end{sagesilent}
\begin{theorem}[Bound on $r$ \#1]
\label{thm:rmax_with_uniform_eps}
	Let $v = (R,C\ell,D\ell^2)$ be a fixed Chern character. Then the ranks of the
	pseudo-semistabilisers for $v$,
	which are solutions to problem \ref{problem:problem-statement-2},
	with $\chern_1^\beta = q$
	are bounded above by the following expression.

	\begin{align*}
		\min
		\left(
			\sage{main_theorem1.r_upper_bound1}, \:\:
			\sage{main_theorem1.r_upper_bound2}
		\right)
	\end{align*}

	Taking the maximum of this expression over
	$q \in [0, \chern_1^{\beta}(F)]\cap \frac{1}{n}\ZZ$
	would give an upper bound for the ranks of all solutions to problem
	\ref{problem:problem-statement-2}.
\end{theorem}

\begin{proof}

\noindent
Both $d$ and the lower bound in
(eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
So, if any of the two upper bounds on $d$ come to within
$\frac{1}{\lcm(m,2n^2)}$ of this lower bound, then there are no solutions for
$d$.
Hence any corresponding $r$ cannot be a rank of a
pseudo-semistabiliser for $v$.

To avoid this, we must have,
considering Equations
\ref{eqn:bgmlv2_d_bound_betamin},
\ref{eqn:bgmlv3_d_bound_betamin},
\ref{eqn:radiuscond_d_bound_betamin}.

\begin{sagesilent}
from plots_and_expressions import \
assymptote_gap_condition1, assymptote_gap_condition2, k
\end{sagesilent}


\begin{align}
	&\sage{assymptote_gap_condition1.subs(k==1)} \\
	&\sage{assymptote_gap_condition2.subs(k==1)}
\end{align}

\noindent
This is equivalent to:

\begin{equation}
	\label{eqn:thm-bound-for-r-impossible-cond-for-r}
	r \leq
	\min\left(
		\sage{
			main_theorem1.r_upper_bound1
		} ,
		\sage{
			main_theorem1.r_upper_bound2
		}
	\right)
\end{equation}

\end{proof}


\begin{sagesilent}
from plots_and_expressions import q_sol, bgmlv_v, psi
\end{sagesilent}

\begin{corollary}[Bound on $r$ \#2]
\label{cor:direct_rmax_with_uniform_eps}
	Let $v$ be a fixed Chern character and
	$R\coloneqq\chern_0(v) \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
	Then the ranks of the pseudo-semistabilisers for $v$,
	which are solutions to problem \ref{problem:problem-statement-2},
	are bounded above by the following expression.

	\begin{equation*}
		\sage{main_theorem1.corollary_r_bound}
	\end{equation*}
\end{corollary}

\begin{proof}
The ranks of the pseudo-semistabilisers for $v$ are bounded above by the
maximum over $q\in [0, \chern_1^{\beta}(F)]$ of the expression in Theorem
\ref{thm:rmax_with_uniform_eps}.
Noticing that the expression is a maximum of two quadratic functions in $q$:
\begin{equation*}
	f_1(q)\coloneqq\sage{main_theorem1.r_upper_bound1} \qquad
	f_2(q)\coloneqq\sage{main_theorem1.r_upper_bound2}
\end{equation*}
These have their minimums at $q=0$ and $q=\chern_1^{\beta}(F)$ respectively.
It suffices to find their intersection in
$q\in [0, \chern_1^{\beta}(F)]$, if it exists,
and evaluating on of the $f_i$ there.
The intersection exists, provided that
$f_1(\chern_1^{\beta}(F)) \geq f_2(\chern_1^{\beta}(F))=R$,
or equivalently,
$R \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
Solving for $f_1(q)=f_2(q)$ yields
\begin{equation*}
	q=\sage{q_sol.expand()}
\end{equation*}
And evaluating $f_1$ at this $q$-value gives:
\begin{equation*}
	\sage{main_theorem1.corollary_intermediate}
\end{equation*}
Finally, noting that $\Delta(v)=\psi^2\ell^2$, we get the bound as
stated in the corollary.

\end{proof}

\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-second}
Just like in example \ref{exmpl:recurring-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{recurring.betaminus}$,
giving $n=\sage{recurring.n}$.

Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilisers for $v$ are bounded above by
$\sage{recurring.corrolary_bound} \approx  \sage{float(recurring.corrolary_bound)}$,
which is much closer to real maximum 25 than the original bound 144.
\end{example}

\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-second}
Just like in example \ref{exmpl:extravagant-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{extravagant.betaminus}$,
giving $n=\sage{extravagant.n}$.

Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilisers for $v$ are bounded above by
$\sage{extravagant.corrolary_bound} \approx  \sage{float(extravagant.corrolary_bound)}$,
which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
original bound 215296.
\end{example}
%% refinements using specific values of q and beta

These bound can be refined a bit more by considering restrictions from the
possible values that $r$ take.
Furthermore, the proof of Theorem \ref{thm:rmax_with_uniform_eps} uses the fact
that, given an element of $\frac{1}{2n^2}\ZZ$, the closest non-equal element of
$\frac{1}{2}\ZZ$ is at least $\frac{1}{2n^2}$ away. However this a
conservative estimate, and a larger gap can sometimes be guaranteed if we know
this value of $\frac{1}{2n^2}\ZZ$ explicitly.

The expressions that will follow will be a bit more complicated and have more
parts which depend on the values of $q$ and $\beta$, even their numerators
$\aa,\bb$ specifically. The upcoming Theorem \ref{thm:rmax_with_eps1} is less useful as a
`clean' formula for a bound on the ranks of the pseudo-semistabilisers, but has a
purpose in the context of writing a computer program to find
pseudo-semistabilisers. Such a program would iterate through possible values of
$q$, then iterate through values of $r$ within the bounds (dependent on $q$),
which would then determine $c$, and then find the corresponding possible values
for $d$.


Firstly, we only need to consider $r$-values for which $c\coloneqq\chern_1(E)$ is
integral:

\begin{equation*}
	c =
	\sage{c_in_terms_of_q.subs([q_value_expr,beta_value_expr])}
	\in \ZZ
\end{equation*}

\noindent
That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
$n$, and so invertible mod $n$).


\noindent
Let $\aa^{'}$ be an integer representative of $\aa^{-1}$ in $\ZZ/n\ZZ$.

Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_F$ in the
proof of Theorem \ref{thm:rmax_with_uniform_eps}:

\begin{lemmadfn}[
	Finding a better alternative to $\epsilon_v$:
	$\epsilon_{v,q}$
	]
	\label{lemdfn:epsilon_q}
	Suppose $d \in \frac{1}{\lcm(m,2n^2)}\ZZ$ satisfies the condition in
	eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
	That is:

	\begin{equation*}
		\sage{positive_radius_condition_with_q.subs([q_value_expr,beta_value_expr]).factor()}
	\end{equation*}

	\noindent
	Then we have:

	\begin{equation}
		\label{eqn:epsilon_q_lemma_prop}
		d - \frac{(\aa r + 2\bb)\aa}{2n^2}
		\geq \epsilon_{v,q} \geq \epsilon_v > 0
	\end{equation}

	\noindent
	Where $\epsilon_{v,q}$ is defined as follows:

	\begin{equation*}
		\epsilon_{v,q} \coloneqq
		\frac{k_{q}}{\lcm(m,2n^2)}
	\end{equation*}
	with $k_{v,q}$ being the least $k\in\ZZ_{>0}$ satisfying
	\begin{equation*}
		k \equiv -\aa\bb \frac{m}{\gcd(m,2n^2)}
		\mod{\gcd\left(
			\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
			\frac{mn\aa}{\gcd(m,2n^2)}
		\right)}
	\end{equation*}
	
\end{lemmadfn}

\vspace{10pt}

\begin{proof}

Consider the following sequence of logical implications.
The one-way implication follows from
$\aa r + \bb \equiv 0 \pmod{n}$,
and the final logical equivalence is just a simplification of the expressions.

\begin{align}
	\frac{ x }{ \lcm(m,2) }
	- \frac{
		(\aa r+2\bb)\aa
	}{
		2n^2
	}
	= \frac{ k }{ \lcm(m,2n^2) }
	\quad \text{for some } x \in \ZZ
	\span \span \span \span \span
	\label{eqn:finding_better_eps_problem}
\\ \nonumber
\\ \Leftrightarrow& &
	- (\aa r+2\bb)\aa
	\frac{\lcm(m,2n^2)}{2n^2}
	&\equiv k &&
	\nonumber
\\ &&&
	\mod \frac{\lcm(m,2n^2)}{\lcm(m,2)}
	\span \span \span
	\nonumber
\\ \Rightarrow& &
	- \bb\aa
	\frac{\lcm(m,2n^2)}{2n^2}
	&\equiv k &&
	\nonumber
\\ &&&
	\mod \gcd\left(
		\frac{\lcm(m,2n^2)}{\lcm(m,2)},
		\frac{n \aa \lcm(m,2n^2)}{2n^2}
	\right)
	\span \span \span
	\nonumber
\\ \Leftrightarrow& &
	- \bb\aa
	\frac{m}{\gcd(m,2n^2)}
	&\equiv k &&
	\label{eqn:better_eps_problem_k_mod_n}
\\ &&&
	\mod \gcd\left(
		\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
		\frac{mn \aa}{\gcd(m,2n^2)}
	\right)
	\span \span \span
	\nonumber
\end{align}

In our situation, we want to find the least $k>0$ satisfying 
eqn \ref{eqn:finding_better_eps_problem}.
Since such a $k$ must also satisfy eqn \ref{eqn:better_eps_problem_k_mod_n},
we can pick the smallest $k_{q,v} \in \ZZ_{>0}$ which satisfies this new condition
(a computation only depending on $q$ and $\beta$, but not $r$).
We are then guaranteed that $k_{v,q}$ is less than any $k$ satisfying eqn
\ref{eqn:finding_better_eps_problem}, giving the first inequality in eqn
\ref{eqn:epsilon_q_lemma_prop}.
Furthermore, $k_{v,q}\geq 1$ gives the second part of the inequality:
$\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.

\end{proof}

\begin{sagesilent}
from plots_and_expressions import main_theorem2
\end{sagesilent}
\begin{theorem}[Bound on $r$ \#3]
\label{thm:rmax_with_eps1}
	Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
	rational and expressed in lowest terms.
	Then the ranks $r$ of the pseudo-semistabilisers $u$ for $v$ with,
	which are solutions to problem \ref{problem:problem-statement-2},
	$\chern_1^\beta(u) = q = \frac{b_q}{n}$
	are bounded above by the following expression:

	\begin{align*}
		\min
		\left(
			\sage{main_theorem2.r_upper_bound1}, \:\:
			\sage{main_theorem2.r_upper_bound2}
		\right)
	\end{align*}
	Where $k_{v,q}$ is defined as in definition/Lemma \ref{lemdfn:epsilon_q},
	and $R = \chern_0(v)$

	Furthermore, if $\aa \not= 0$ then
	$r \equiv \aa^{-1}b_q \pmod{n}$.
\end{theorem}

Although the general form of this bound is quite complicated, it does simplify a
lot when $m$ is small.

\begin{sagesilent}
from plots_and_expressions import main_theorem2_corollary
\end{sagesilent}
\begin{corollary}[Bound on $r$ \#3 on $\PP^2$ and Principally polarized abelian surfaces]
\label{cor:rmax_with_eps1}
	Suppose we are working over $\PP^2$ or a principally polarized abelian surface
	(or any other surfaces with $m=1$ or $2$).
	Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
	rational and expressed in lowest terms.
	Then the ranks $r$ of the pseudo-semistabilisers $u$ for $v$ with,
	which are solutions to problem \ref{problem:problem-statement-2},
	$\chern_1^\beta(u) = q = \frac{b_q}{n}$
	are bounded above by the following expression:

	\begin{align*}
		\min
		\left(
			\sage{main_theorem2_corollary.r_upper_bound1}, \:\:
			\sage{main_theorem2_corollary.r_upper_bound2}
		\right)
	\end{align*}
	Where $R = \chern_0(v)$ and $k_{v,q}$ is the least
	$k\in\ZZ_{>0}$ satisfying
	\begin{equation*}
		k \equiv -\aa\bb
		\pmod{n}
	\end{equation*}

	\noindent
	Furthermore, if $\aa \not= 0$ then
	$r \equiv \aa^{-1}b_q \pmod{n}$.
\end{corollary}

\begin{proof}
This is a specialisation of Theorem \ref{thm:rmax_with_eps1}, where we can
drastically simplify the $\lcm$ and $\gcd$ terms by noting that $m$ divides both
$2$ and $2n^2$, and that $a_v$ is coprime to $n$.
\end{proof}

\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-third}
Just like in examples \ref{exmpl:recurring-first} and
\ref{exmpl:recurring-second},
take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
$\beta=\sage{recurring.betaminus}$, giving $n=\sage{recurring.n}$
and $\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
%% TODO transcode notebook code
The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabiliser $u$ of $v$
in terms of the possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:

\begin{sagesilent}
from examples import bound_comparisons
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
\end{sagesilent}

\vspace{1em}
\noindent
\directlua{ table_width = 3*4+1 }
\begin{tabular}{l\directlua{for i=0,table_width-1 do tex.sprint([[|c]]) end}}
	$q=\chern_1^\beta(u)$
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	\\ \hline
	Thm \ref{thm:rmax_with_uniform_eps}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	\\
	Thm \ref{thm:rmax_with_eps1}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
\end{tabular}
\vspace{1em}

\noindent
It's worth noting that the bounds given by Theorem \ref{thm:rmax_with_eps1}
reach, but do not exceed the actual maximum rank 25 of the
pseudo-semistabilisers of $v$ in this case.
As a reminder, the original loose bound from Theorem \ref{thm:loose-bound-on-r}
was 144.

\end{example}

\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-third}
Just like in examples \ref{exmpl:extravagant-first} and
\ref{exmpl:extravagant-second},
take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
$\beta=\sage{extravagant.betaminus}$, giving $n=\sage{n:=extravagant.n}$
and $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
This example was chosen because the $n$ value is moderatly large, giving more
possible values for $k_{v,q}$, in dfn/Lemma \ref{lemdfn:epsilon_q}. This allows
for a larger possible difference between the bounds given by Theorems
\ref{thm:rmax_with_uniform_eps} and \ref{thm:rmax_with_eps1}, with the bound
from the second being up to $\sage{n}$ times smaller, for any given $q$ value.
The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabiliser $u$ of $v$
in terms of the first few smallest possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:

\begin{sagesilent}
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(extravagant)
\end{sagesilent}


\vspace{1em}
\noindent
\directlua{ table_width = 12 }
\begin{tabular}{l\directlua{for i=0,table_width do tex.sprint([[|c]]) end}}
	$q=\chern_1^\beta(u)$
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	&$\cdots$
	\\ \hline
	Thm \ref{thm:rmax_with_uniform_eps}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	&$\cdots$
	\\
	Thm \ref{thm:rmax_with_eps1}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	&$\cdots$
\end{tabular}
\vspace{1em}


\noindent
However the reduction in the overall bound on $r$ is not as drastic, since all
possible values for $k_{v,q}$ in $\{1,2,\ldots,\sage{n}\}$ are iterated through
cyclically as we consider successive possible values for $q$.
And for each $q$ where $k_{v,q}=1$, both Theorems give the same bound.
Calculating the maximums over all values of $q$ yields
$\sage{max(theorem2_bounds)}$ for Theorem \ref{thm:rmax_with_uniform_eps}, and
$\sage{max(theorem3_bounds)}$ for Theorem \ref{thm:rmax_with_eps1}.
\end{example}

\egroup % end scope where beta redefined to beta_{-}