From d617f6b99e6e1a2bdca898c3b0693e206034a5d6 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Mon, 24 Jul 2023 17:18:55 +0100
Subject: [PATCH] Start adjusting references in beginning of section with
theorems
---
main.tex | 33 +++++++++++++++------------------
1 file changed, 15 insertions(+), 18 deletions(-)
diff --git a/main.tex b/main.tex
index 14e8b49..80fc261 100644
--- a/main.tex
+++ b/main.tex
@@ -761,6 +761,7 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
\section{B.Schmidt's Solutions to the Problems}
\subsection{Bound on $\chern_0(u)$ for Semistabilizers}
+\label{subsect:loose-bound-on-r}
The proof for the following theorem \ref{thm:loose-bound-on-r} was hinted at in
\cite{SchmidtBenjamin2020Bsot}, but the value appears explicitly in
@@ -1288,6 +1289,7 @@ previous subsubsections.
\subsubsection{All Bounds on $d$ Together for Problem
\ref{problem:problem-statement-2}}
+\label{subsubsect:all-bounds-on-d-prob2}
%% RECAP ON INEQUALITIES TOGETHER
%%%% RATIONAL BETA MINUS
@@ -1498,23 +1500,16 @@ Some of the details around the associated numerics are explored next.
\label{fig:d_bounds_xmpl_gnrc_q}
\end{figure}
-\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{r}}
+\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{r} in Problem
+\ref{problem:problem-statement-2}}
-Now, the inequalities from the above (TODO REF) will be used to find, for
+Now, the inequalities from the above subsubsection
+\ref{subsubsect:all-bounds-on-d-prob2} 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 $V_v$ for Rational beta}
-
-
-The strategy here is similar to what was shown in (sect
-\ref{sec:twisted-chern}).
-One specialization here is to use that $\ell\coloneqq c_1(H)$ 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?
+no possible solutions for $d$. At that point, there are no solutions
+$u=(r,c\ell,d\ell^2)$ to problem \ref{problem:problem-statement-2}.
+The strategy here is similar to what was shown in theorem
+\ref{thm:loose-bound-on-r}.
\begin{sagesilent}
var("a_v b_q n") # Define symbols introduce for values of beta and q
@@ -1533,11 +1528,13 @@ Then fix a value of $q$:
\frac{1}{n} \ZZ
\cap [0, \chern_1^{\beta}(F)]
\end{equation}
-as noted at the beginning of this section (\ref{sec:refinement}).
+as noted at the beginning of this section \ref{sec:refinement} so that we are
+considering $u$ which satisfy \ref{item:chern1bound:lem:num_test_prob2}
+in corollary \ref{cor:num_test_prob2}.
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:
+(eqn \ref{eqn:radiuscond_d_bound_betamin}) we get:
\begin{equation}
\label{eqn:positive_rad_condition_in_terms_of_q_beta}
@@ -1611,7 +1608,7 @@ 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}.
+\ref{eqn:radiuscond_d_bound_betamin}.
\bgroup
--
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