main.tex

\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 $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ 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 $q$.
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}

\section{Computing solutions to Problem \ref{problem:problem-statement-2}}
\label{sect:prob2-algorithm}

Alongside this article, there is a library \cite{NaylorRust2023} to compute
the solutions to problem \ref{problem:problem-statement-2}, using the theorems
above.

The way it works, is by yielding solutions to the problem
$u=(r,c\ell,\frac{e}{2}\ell^2)$ as follows.

\subsection{Iterating Over Possible $q=\chern^{\beta_{-}}(u)$}

Given a Chern character $v$, the domain of the problem are first verified: that
$v$ has positive rank, that it satisfies $\Delta(v) \geq 0$, and that
$\beta_{-}(v)$ is rational.

Take $\beta_{-}(v)=\frac{a_v}{n}$ in simplest terms.
Iterate over $q \in [0,\chern_1^{\beta_{-}}(v)]\cap\frac{1}{n}\ZZ$.

For any $u = (r,c\ell,\frac{e}{2}\ell^2)$, satisfying
$\chern_1^{\beta_{-}}(u)=q$ for one of the $q$ considered is equivalent to
satisfying condition \ref{item:chern1bound:lem:num_test_prob2}
in corollary \ref{cor:num_test_prob2}.

\subsection{Iterating Over Possible $r=\chern_0(u)$ for Fixed $q=\chern^{\beta_{-}}(u)$}

Let $q=\frac{b_q}{n}$ be one of the values of $\chern_1^{\beta_{-}}(u)$ that we
have fixed. As mentioned before, the only values of $r$ which can
give $\chern_1^{\beta_{-}}(u)=q$ are precisely the ones which satisfy
$a_v r \equiv b_q \pmod{n}$.
This is true for all integers when $\beta_{-}=0$ (and so $n=1$), but otherwise,
this is equivalent to
$r \equiv {a_v}^{-1}b_q \pmod{n}$, since $a_v$ and $n$ are coprime.

Note that expressing $\mu(u)$ in term of $q$ and $r$ gives:
\begin{align*}
	\mu(u) & = \frac{c}{r} = \frac{q+r\beta_{-}}{r}
	\\
	&= \beta_{-} + \frac{q}{r}
\end{align*}

So condition \ref{item:mubound:lem:num_test_prob2} in corollary
\ref{cor:num_test_prob2} is satisfied at this point precisely when:

\begin{equation*}
	r > \frac{q}{\mu(u) - \beta_{-}}
\end{equation*}

Note that the right hand-side is greater than, or equal, to 0, so such $r$ also
satisfies \ref{item:rankpos:lem:num_test_prob2}.

Then theorem \ref{thm:rmax_with_eps1} gives an upper on possible $r$ values
for which it is possible to satisfy conditions
\ref{item:bgmlvu:lem:num_test_prob2},
\ref{item:bgmlvv-u:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob2}.

Iterate over such $r$ so that we are guarenteed to satisfy conditions
\ref{item:mubound:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob2}
in corollary
\ref{cor:num_test_prob2}, and have a chance at satisfying the rest.

\subsection{Iterating Over Possible $d=\chern_2(u)$ for Fixed $r=\chern_0(u)$
and $q=\chern^{\beta_{-}}(u)$}

At this point we have fixed $\chern_0(u)=r$ and
$\chern_1(u)=c=q+r\beta_{-}$.
And the cases considered are precisely the ones which satisfy conditions
\ref{item:chern1bound:lem:num_test_prob2},
\ref{item:mubound:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob2}
in corollary \ref{cor:num_test_prob2}.

It remains to find $\chern_2(u)=d=\frac{e}{2}$
which satisfy the remaining conditions
\ref{item:bgmlvu:lem:num_test_prob2},
\ref{item:bgmlvv-u:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob2}.
These conditions induce upper and lower bounds on $d$, and it then remains to
just pick the integers $e$ that give $d$ values within the bounds.

Thus, through this process yielding all solutions $u=(r,c\ell,\frac{e}{2}\ell^2)$
to the problem for this choice of $v$.



\newpage
\printbibliography

\newpage
\section{Appendix - SageMath code}

\usemintedstyle{tango}

\begin{footnotesize}
\inputminted[
	obeytabs=true,
	tabsize=2,
	breaklines=true,
	breakbefore=./
]{python}{filtered_sage.txt}
\end{footnotesize}

\end{document}