From 53ffe34442cb6127cb909aa0c0e651e3e62ebac9 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 3 Jul 2024 17:55:38 +0100
Subject: [PATCH] Sort out statements and introduction leading up to bounds on
d
---
tex/bounds-on-semistabilisers.tex | 63 +++++++++++++++++++------------
1 file changed, 38 insertions(+), 25 deletions(-)
diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index 3239093..b536737 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -161,27 +161,8 @@ corresponding $\chern_1^{\beta}(u)$ fail one of the inequalities (which is what
was implicitly happening before in the proof of Theorem
\ref{thm:loose-bound-on-r}).
-
-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 $\beta_0 = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
-\ref{problem:problem-statement-1}
-(or $\beta_0 = \beta_{-}$ 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}
-
\begin{lemma}
+\label{lem:fixed-q-semistabs-criterion}
Consider the Problem \ref{problem:problem-statement-1} (or \ref{problem:problem-statement-2}),
and $\beta_{0}\coloneqq \beta(P)$ (or $\beta_{-}(v)$ resp.).
@@ -214,6 +195,7 @@ Take $\beta_0 = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
\end{proof}
\begin{corollary}
+\label{cor:rational-beta:fixed-q-semistabs-criterion}
Consider the Problem \ref{problem:problem-statement-1} (or \ref{problem:problem-statement-2}),
and suppose that $\beta_{0}\coloneqq \beta(P)$ (or $\beta_{-}(v)$ resp.) is
rational, and written $\beta_0=\frac{a_v}{n}$ for
@@ -248,11 +230,40 @@ Take $\beta_0 = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
\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 $d$.
+Let $v$ be a Chern character with $\Delta(v)\geq 0$,
+$\chern_0(v)>0$, or $\chern_0(v)=0$ and $\chern_1(v)>0$,
+and consider Problem
+\ref{problem:problem-statement-1} or
+\ref{problem:problem-statement-2}.
+Take $\beta_0 = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
+\ref{problem:problem-statement-1}
+(or $\beta_0 = \beta_{-}$ for problem \ref{problem:problem-statement-2}).
+Lemma \ref{lem:fixed-q-semistabs-criterion} states that any solution
+\[
+ u \coloneqq \:(r,c\ell,d\ell^2)
+ \qquad
+ \text{where $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$}
+ \label{eqn:u-coords}
+\]
+to the Problem satisfies
+\[
+ q \coloneqq \chern_1^{\beta_0}(u)
+ \in
+ \left(
+ 0, \chern_1^{\beta_0}(v)
+ \right)
+\]
+and also gives a lower bound for $r$ when considering $u$ with a fixed $q$.
+This Section studies the extra numerical conditions that such $u$ must satisfy
+as per that Lemma, and will express them as bounds on $d$ in terms of $r$ (for a
+fixed $q$).
+These bounds will later be used in Subsections
+\ref{subsec:bounds-on-semistab-rank-prob-1} and
+\ref{subsec:bounds-on-semistab-rank-prob-2}
+to construct upper bounds on $r$
+(in a similar way to how a bound on $\chern_0(u)$ was found in the proof of
+Theorem \ref{thm:loose-bound-on-r} by considering bounds on
+$\chern^{\beta_0}_0(u)$ in terms of the former).
\subsubsection{Size of pseudo-wall\texorpdfstring{: $\chern_2^P(u)>0$}{}}
\label{subsect-d-bound-radiuscond}
@@ -594,6 +605,7 @@ A generic example of this is plotted in Figure
\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{} in Problem
\ref{problem:problem-statement-1}}
+\label{subsec:bounds-on-semistab-rank-prob-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}),
@@ -671,6 +683,7 @@ following Lemma \ref{lem:prob1:convenient_r_bound}.
\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{} in Problem
\ref{problem:problem-statement-2}}
+\label{subsec:bounds-on-semistab-rank-prob-2}
Now, the inequalities from the above subsubsection
\ref{subsubsect:all-bounds-on-d-prob2} will be used to find, for
--
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