From 22689d8b39aabe89debb78242ff5d09895ec121c Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu, 22 Jun 2023 16:46:53 +0100
Subject: [PATCH] Add extravagant set of examples
---
main.tex | 108 ++++++++++++++++++++++++++++++++++++++++++++++++++++++-
1 file changed, 107 insertions(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index f26a6c4..6eb1e74 100644
--- a/main.tex
+++ b/main.tex
@@ -548,6 +548,33 @@ 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{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
+\label{exmpl:extravagant-first}
+\begin{sagesilent}
+extravagant = Object()
+extravagant.chern = Chern_Char(29, 13, -3/2)
+extravagant.b = beta_minus(extravagant.chern)
+extravagant.twisted = extravagant.chern.twist(extravagant.b)
+extravagant.actual_rmax = 49313
+\end{sagesilent}
+Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
+that $m=2$, $\beta_-=\sage{extravagant.b}$,
+giving $n=\sage{extravagant.b.denominator()}$ and
+$\chern_1^{\sage{extravagant.b}}(F) = \sage{extravagant.twisted.ch[1]}$.
+
+\begin{sagesilent}
+n = extravagant.b.denominator()
+m = 2
+loose_bound = (
+ m*n^2*extravagant.twisted.ch[1]^2
+) / gcd(m, 2*n^2)
+\end{sagesilent}
+Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
+tilt semistabilizers for $v$ are bounded above by $\sage{loose_bound}$.
+However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
+rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
+\end{example}
+
\section{B.Schmidt's Method}
Goals:
@@ -1498,6 +1525,29 @@ 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 7744.
+\end{example}
%% refinements using specific values of q and beta
These bound can be refined a bit more by considering restrictions from the
@@ -1710,8 +1760,9 @@ def bound_comparisons(example):
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
\end{sagesilent}
+\noindent
\directlua{ table_width = 3*4+1 }
-\directlua{ table_width = 3*4+1 }\begin{tabular}{l\directlua{for i=0,table_width-1 do tex.sprint([[|c]]) end}}
+\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 = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
@@ -1739,6 +1790,61 @@ 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}
+
+
+\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}
+
+
+\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
--
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