main.tex

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}

\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}