main.tex

	}{
		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
$\sage{bgmlv3_d_upperbound_linear_term}$.
Finally, there is an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
and can be written:
\bgroup
\def\psi{\chern_1^{\beta}(F)}
$\sage{bgmlv3_d_upperbound_exp_term_alt2}$.
\egroup
In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
$\chern^{\beta}_2(F) = 0$,
so some of these expressions simplify, and in particular, the constant and
linear terms match those of the other bounds in the previous subsections.

\subsubsection{All Bounds on $d$ together}
%% RECAP ON INEQUALITIES TOGETHER

%%%% RATIONAL BETA MINUS
\minorheading{Special Case: Rational $\beta_{-}$}

Suppose we take $\beta = \beta_{-}$ (so that $\chern^{\beta}_2(F)=0$)
in the previous subsections, to find all circular walls to the left of the
vertical wall (TODO as discussed in ref).

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

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

\bgroup
% redefine \psi in sage expressions (placeholder for ch_1^\beta(F)
\def\psi{\chern_1^{\beta}(F)}
\begin{align}
	d &\geq&
	\sage{bgmlv1_d_lowerbound_linear_term}
	&+ \sage{bgmlv1_d_lowerbound_const_term_alt.subs(chbv == 0)}
	+& \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)},
	 &\qquad\text{when\:} r > \frac{R}{2}
	 \label{eqn:bgmlv1_d_bound_betamin}
\\
	d &\leq&
	\sage{bgmlv2_d_upperbound_linear_term}
	&+ \sage{bgmlv2_d_upperbound_const_term}
	+& \sage{bgmlv2_d_upperbound_exp_term},
	 &\qquad\text{when\:} r > 0
	 \label{eqn:bgmlv2_d_bound_betamin}
\\
	d &\leq&
	\sage{bgmlv3_d_upperbound_linear_term}
	&+ \sage{bgmlv3_d_upperbound_const_term_alt.subs(phi == 0)}
	% ^ ch_2^\beta(F)=0 for beta_{-}
	+& \sage{bgmlv3_d_upperbound_exp_term_alt2},
	 &\qquad\text{when\:} r > R
	 \label{eqn:bgmlv3_d_bound_betamin}
\end{align}
\egroup

Furthermore, we get an extra bound for $d$ resulting from the condition that the
radius of the circular wall must be positive. As discussed in (TODO ref), this
is equivalent to $\chern^{\beta}_2(E) > 0$, which yields:

\begin{sagesilent}
positive_radius_condition = (
	(
		(0 > - u.twist(beta).ch[2])
		+ d # rearrange for d
	)
	.subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q
	.expand()
)
\end{sagesilent}

\begin{equation}
	\label{eqn:positive_rad_d_bound_betamin}
	\sage{positive_radius_condition}
\end{equation}
\begin{sagesilent}
def beta_min(chern):
  ts = stability.Tilt()
  return min(
    map(
      lambda soln: soln.rhs(),
      solve(
        (ts.degree(chern))
          .expand()
          .subs(ts.alpha == 0),
        beta
      )
    )
  )

v_example = Chern_Char(3,2,-2)
q_example = 7/3

def plot_d_bound(
  v_example,
  q_example,
  ymax=5,
  ymin=-2,
  xmax=20,
  aspect_ratio=None
):

  # Equations to plot imminently representing the bounds on d:
  eq1 = (
    bgmlv1_d_lowerbound
    .subs(R == v_example.ch[0])
    .subs(C == v_example.ch[1])
    .subs(D == v_example.ch[2])
    .subs(beta = beta_min(v_example))
    .subs(q == q_example)
  )

  eq2 = (
    bgmlv2_d_upperbound
    .subs(R == v_example.ch[0])
    .subs(C == v_example.ch[1])
    .subs(D == v_example.ch[2])
    .subs(beta = beta_min(v_example))
    .subs(q == q_example)
  )

  eq3 = (
    bgmlv3_d_upperbound
    .subs(R == v_example.ch[0])
    .subs(C == v_example.ch[1])
    .subs(D == v_example.ch[2])
    .subs(beta = beta_min(v_example))
    .subs(q == q_example)
  )

  eq4 = (
    positive_radius_condition.rhs()
    .subs(q == q_example)
    .subs(beta = beta_min(v_example))
  )

  example_bounds_on_d_plot = (
    plot(
      eq3,
      (r,v_example.ch[0],xmax),
      color='green',
			linestyle = "dashed",
      legend_label=r"upper bound: $\Delta(G) \geq 0$",
    )
    + plot(
      eq2,
      (r,0,xmax),
      color='blue',
			linestyle = "dashed",
      legend_label=r"upper bound: $\Delta(E) \geq 0$"
    )
    + plot(
      eq4,
      (r,0,xmax),
      color='orange',
			linestyle = "dotted",
      legend_label=r"lower bound: $\mathrm{ch}_2^{\beta_{-}}(E)>0$"
    )
    + plot(
      eq1,
      (r,v_example.ch[0]/2,xmax),
      color='red',
			linestyle = "dotted",
      legend_label=r"lower bound: $\Delta(E) + \Delta(G) \leq \Delta(F)$"
    )
  )
  example_bounds_on_d_plot.ymin(ymin)
  example_bounds_on_d_plot.ymax(ymax)
  example_bounds_on_d_plot.axes_labels(['$r$', '$d$'])
  if aspect_ratio:
    example_bounds_on_d_plot.set_aspect_ratio(aspect_ratio)
  return example_bounds_on_d_plot

\end{sagesilent}

\begin{figure}
\centering
\begin{subfigure}{.45\textwidth}
  \centering
	\sageplot[width=\linewidth]{plot_d_bound(v_example, 0)}
	\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]{plot_d_bound(v_example, 4)}
	\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 $d:=\chern_2(E)$ in terms of $r:=\chern_0(E)$ for fixed, extreme,
	values of $q:=\chern_1^{\beta}(E)$.
	Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_extrm_q}
\end{figure}

Recalling that $q := \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
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 destabilizers $E$ with
$\chern_0(E)=:r>R:=\chern_0(F)$ 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:positive_rad_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:positive_rad_d_bound_betamin}) which match.


The more generic case, when $0 < q:=\chern_1^{\beta}(E) < \chern_1^{\beta}(F)$
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-destabilizers of arbitrarily large
rank, in the context of the graph, comes down to whether there are points
$(r,d) \in \ZZ \oplus \frac{1}{2} \ZZ$ (with large $r$)
% TODO have a proper definition for pseudo-destabilizers/walls
that fit above the yellow line (ensuring positive radius of wall) but below the
blue and green (ensuring $\Delta(E), \Delta(G) > 0$).
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:positive_rad_d_bound_betamin}).
As mentioned in the introduction (sec \ref{sec:intro}), 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{figure}
\centering
\sageplot[
	width=\linewidth
]{plot_d_bound(v_example, 2, ymax=6, ymin=-0.5, aspect_ratio=1)}
\caption{
	Bounds on $d:=\chern_2(E)$ in terms of $r:=\chern_0(E)$ for a fixed
	value $\chern_1^{\beta}(F)/2$ of $q:=\chern_1^{\beta}(E)$.
	Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_gnrc_q}
\end{figure}

\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{r}}

Now, the inequalities from the above (TODO REF) 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 Chern characters
$(r,c,d)$ that satisfy all inequalities to give a pseudowall.


\subsubsection{All Semistabilizers Left of Vertical Wall for Rational Beta min}


The strategy here is similar to what was shown in (sect
\ref{sec:twisted-chern}).
One specialization here is to use that $\ell:=c_1(H)$ generates $NS(X)$, so that
in fact, any Chern character can be written as
$\left(r,c\ell,\frac{e}{2}\ell^2\right)$ for $r,c,e\in\ZZ$.
% ref to Schmidt?

\begin{sagesilent}
var("a_v b_q n") # Define symbols introduce for values of beta and q
beta_value_expr = (beta == a_v/n)
q_value_expr = (q == b_q/n)
\end{sagesilent}

\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:=\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}).

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

\begin{equation}
\label{eqn:positive_rad_condition_in_terms_of_q_beta}
	\frac{1}{2}\ZZ
	\ni
	\qquad
	\sage{positive_radius_condition.subs([q_value_expr,beta_value_expr]).factor()}
	\qquad
	\in
	\frac{1}{2n^2}\ZZ
\end{equation}

\begin{sagesilent}
# placeholder for the specific values of k (start with 1):
var("kappa", domain="real")

assymptote_gap_condition1 = (kappa/(2*n^2) < bgmlv2_d_upperbound_exp_term)
assymptote_gap_condition2 = (kappa/(2*n^2) < bgmlv3_d_upperbound_exp_term_alt2)

r_upper_bound1 = (
	assymptote_gap_condition1
	* r * 2*n^2 / kappa
)

assert r_upper_bound1.lhs() == r

r_upper_bound2 = (
	assymptote_gap_condition2
	* (r-R) * 2*n^2 / kappa + R
)

assert r_upper_bound2.lhs() == r
\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-semistabilizers for $v$ with
	$\chern_1^\beta = q$
	are bounded above by the following expression.

	\bgroup
	\def\psi{\chern_1^{\beta}(F)}
	\begin{align*}
		\min
		\left(
			\sage{r_upper_bound1.subs(kappa==1).rhs()}, \:\:
			\sage{r_upper_bound2.subs(kappa==1).rhs()}
		\right)
	\end{align*}
	\egroup

	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 pseudo-semistabilizers for $v$.
\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 $\frac{1}{2n^2}\ZZ$.
So, if any of the two upper bounds on $d$ come to within
$\frac{1}{2n^2}$ of this lower bound, then there are no solutions for
$d$.
Hence any corresponding $r$ cannot be a rank of a
pseudo-semistabilizer for $v$.

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

\bgroup

\let\originalepsilon\epsilon
\renewcommand\epsilon{{\originalepsilon_{v}}}

\begin{sagesilent}
var("epsilon")

# Tightness conditions:

bounds_too_tight_condition1 = (
	bgmlv2_d_upperbound_exp_term
	< epsilon
)

bounds_too_tight_condition2 = (
	bgmlv3_d_upperbound_exp_term_alt.subs(chbv==0)
	< epsilon
)
\end{sagesilent}

\bgroup
\def\psi{\chern_1^{\beta}(F)}
\begin{equation}
	\min\left(
		\sage{bgmlv2_d_upperbound_exp_term},
		\sage{bgmlv3_d_upperbound_exp_term_alt2}
	\right)
	\geq \epsilon := \frac{1}{2n^2}
\end{equation}
\egroup

\noindent
This is equivalent to:

\bgroup
\def\psi{\chern_1^{\beta}(F)}
\begin{equation}
	\label{eqn:thm-bound-for-r-impossible-cond-for-r}
	r \leq
	\min\left(
		\sage{
			r_upper_bound1.subs(kappa==1).rhs()
		} ,
		\sage{
			r_upper_bound2.subs(kappa==1).rhs()
		}
	\right)
\end{equation}
\egroup

\egroup % end scope where epsilon redefined

\end{proof}

\begin{sagesilent}
var("Delta nu", domain="real")
q_sol = solve(
  r_upper_bound1.subs(kappa==1).rhs()
  == r_upper_bound2.subs(kappa==1).rhs()
, q)[0].rhs()
r_upper_bound_all_q = (
	r_upper_bound1.rhs()
	.expand()
	.subs(q==q_sol)
	.subs(kappa==1)
	.subs(psi**2 == Delta/nu^2)
	.subs(1/psi**2 == nu^2/Delta)
)
\end{sagesilent}

\begin{corrolary}[Bound on $r$ \#2]
\label{cor:direct_rmax_with_uniform_eps}
	Let $v$ be a fixed Chern character and
	$R:=\chern_0(v) \leq n^2\Delta(v)$.
	Then the ranks of the pseudo-semistabilizers for $v$
	are bounded above by the following expression.

	\bgroup
	\let\originalDelta\Delta
	\let\nu\ell
	\renewcommand\Delta{{\originalDelta(v)}}
	\begin{equation*}
		\sage{r_upper_bound_all_q.expand()}
	\end{equation*}
	\egroup
\end{corrolary}

\begin{proof}
\bgroup
\def\psi{\chern_1^{\beta}(F)}
\let\originalDelta\Delta
The ranks of the pseudo-semistabilizers 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):=\sage{r_upper_bound1.subs(kappa==1).rhs()} \qquad
	f_2(q):=\sage{r_upper_bound2.subs(kappa==1).rhs()}
\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
$f_1(\chern_1^{\beta}(F))>f_2(\chern_1^{\beta}(F))=R$,
or equivalently,
$R \leq n^2{\chern_1^{\beta}(F)}^2$.
Setting $f_1(q)=f_2(q)$ yields
$q=\sage{q_sol.expand()}$.
And evaluating $f_1$ at this $q$-value gives:
$\sage{r_upper_bound_all_q.expand().subs([nu==1,Delta==psi^2])}$.
Finally, noting that $\originalDelta(v)=\psi^2\ell^2$, we get the bound as
stated in the corollary.
\egroup
\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 $m=2$, $\beta=\sage{recurring.b}$,
giving $n=\sage{recurring.b.denominator()}$.

\begin{sagesilent}
recurring.n = recurring.b.denominator()
recurring.bgmlv = recurring.chern.Q_tilt()
corrolary_bound = (
  r_upper_bound_all_q.expand()
  .subs(Delta==recurring.bgmlv)
  .subs(nu==1) ## \ell^2=1 on P^2
  .subs(R==recurring.chern.ch[0])
  .subs(n==recurring.n)
)
\end{sagesilent}
Using the above corrolary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilizers for $v$ are bounded above by
$\sage{corrolary_bound} \approx  \sage{float(corrolary_bound)}$,
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 $m=2$, $\beta=\sage{extravagant.b}$,
giving $n=\sage{extravagant.b.denominator()}$.

\begin{sagesilent}
extravagant.n = extravagant.b.denominator()
extravagant.bgmlv = extravagant.chern.Q_tilt()
corrolary_bound = (
  r_upper_bound_all_q.expand()
  .subs(Delta==extravagant.bgmlv)
  .subs(nu==1) ## \ell^2=1 on P^2
  .subs(R==extravagant.chern.ch[0])
  .subs(n==extravagant.n)
)
\end{sagesilent}
Using the above corrolary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilizers for $v$ are bounded above by
$\sage{corrolary_bound} \approx  \sage{float(corrolary_bound)}$,
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 (TODO ref) is less useful as a
`clean' formula for a bound on the ranks of the pseudo-semistabilizers, but has a
purpose in the context of writing a computer program to find
pseudo-semistabilizers. 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 $c:=\chern_1(E)$ 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 $n$).

\begin{sagesilent}
	rhs_numerator = (
	positive_radius_condition
	.rhs()
	.subs([q_value_expr,beta_value_expr])
	.factor()
	.numerator()
	)
\end{sagesilent}

\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 $d \in \frac{1}{2}\ZZ$ satisfies the condition in
	eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
	That is:

	\begin{equation*}
		\sage{positive_radius_condition.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} :=
		\frac{k_{q}}{2n^2}
	\end{equation*}
	with $k_{v,q}$ being the least $k\in\ZZ_{>0}$ satisfying $k \equiv -\aa\bb \mod n$
	
\end{lemmadfn}

\begin{proof}

Consider the following:

\begin{align}
	\frac{ x }{ 2 }
	- \frac{
		(\aa r+2\bb)\aa
	}{
		2n^2
	}
	= \frac{ k }{ 2n^2 }
	\quad \text{for some } x \in \ZZ
	\span \span \span \span \span
	\label{eqn:finding_better_eps_problem}
\\ \Longleftrightarrow& &
	- (\aa r+2\bb)\aa
	&\equiv k &&
	\mod n^2
\\ \Longleftrightarrow& &
	- \aa^2 r - 2\aa\bb
	&\equiv k &&
	\mod n^2
\\  \Longrightarrow& &
  \aa^2 \aa^{-1}\bb - 2\aa\bb
	&\equiv k &&
	\mod n
	\label{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
\\ \Longleftrightarrow& &
  -\aa\bb
	&\equiv k &&
	\mod n
	\label{eqn:better_eps_problem_k_mod_n}
\end{align}

In our situation, we want to find the least $k$ 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 $k_{q,1} \in \ZZ_{>0}$ 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{theorem}[Bound on $r$ \#3]
\label{thm:rmax_with_eps1}
	Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta:=\beta(v)$
	rational and expressed in lowest terms.
	Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with
	$\chern_1^\beta(u) = q = \frac{b_q}{n}$
	are bounded above by the following expression:

\begin{sagesilent}
var("delta", domain="real") # placeholder symbol to be replaced by k_{q,i}
\end{sagesilent}

	\bgroup
	\def\kappa{k_{v,q}}
	\def\psi{\chern_1^{\beta}(F)}
	\begin{align*}
		\min
		\left(
			\sage{r_upper_bound1.rhs()}, \:\:
			\sage{r_upper_bound2.rhs()}
		\right)
	\end{align*}
	\egroup
	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
	$r \equiv \aa^{-1}b_q (\mod n)$.
\end{theorem}


\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
$\beta=\sage{recurring.b}$, giving $n=\sage{recurring.b.denominator()}$
and $\chern_1^{\sage{recurring.b}}(F) = \sage{recurring.twisted.ch[1]}$.
%% TODO transcode notebook code
The (non-exclusive) upper bounds for $r:=\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
in terms of the possible values for $q:=\chern_1^{\beta}(u)$ are as follows:

\begin{sagesilent}
import numpy as np

def bound_comparisons(example):
    n = example.b.denominator()
    a_v = example.b.numerator()

    def theorem_bound(v_twisted, q_val, k):
      return int(min(
        n^2*q_val^2/k
      ,
        v_twisted.ch[0]
        + n^2*(v_twisted.ch[1] - q_val)^2/k
      ))

    def k(n, a_v, b_q):
      n = int(n)
      a_v = int(a_v)
      b_q = int(b_q)
      k = -a_v*b_q % n
      return k if k > 0 else k + n

    b_qs = list(range(example.twisted.ch[1]*n+1))
    qs = list(map(lambda x: x/n,b_qs))
    ks = list(map(lambda b_q: k(n, a_v, b_q), b_qs))
    theorem2_bounds = [
        theorem_bound(example.twisted, q_val, 1)
        for q_val in qs
    ]
    theorem3_bounds = [
        theorem_bound(example.twisted, q_val, k)
        for q_val, k in zip(qs,ks)
    ]
    return qs, theorem2_bounds, theorem3_bounds

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-semistabilizers 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
$\beta=\sage{extravagant.b}$, giving $n=\sage{n:=extravagant.b.denominator()}$
and $\chern_1^{\sage{extravagant.b}}(F) = \sage{extravagant.twisted.ch[1]}$.
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 $\sage{n}$ smaller, for any given $q$ value.
The (non-exclusive) upper bounds for $r:=\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
in terms of the first few smallest possible values for $q:=\chern_1^{\beta}(u)$ 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 $q$.
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_{-}

\subsubsection{All Semistabilizers Giving Sufficiently Large Circular Walls Left
of Vertical Wall}


Goals:
\begin{itemize}
	\item refresher on strategy
	\item point out no need for rational beta
	\item calculate intersection of bounds?
\end{itemize}

\subsection{Irrational $\beta_{-}$}

Goals:
\begin{itemize}
	\item Point out if only looking for sufficiently large wall, look at above
		subsubsection
	\item Relate to Pell's equation through coordinate change?
	\item Relate to numerical condition described by Yanagida/Yoshioka
\end{itemize}

\newpage
\section{Appendix - SageMath code}

\usemintedstyle{tango}