From 5097f3edb90b28be975255fed7f679b27ac66849 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 3 Jul 2024 23:05:41 +0100
Subject: [PATCH] Start thinking about gap for 3rd condition when r<R
---
tex/bounds-on-semistabilisers.tex | 23 ++++++++++-------------
1 file changed, 10 insertions(+), 13 deletions(-)
diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index 96cadd0..b967fa2 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -288,9 +288,10 @@ $P=(\alpha_0,\beta_0) \in \Theta_v^-$.
Then $\sage{problem1.A2_subs}$ follows from $\chern_2^P(v)=0$. Using this substitution into the
condition $\chern_2^P(u) = \sage{problem1.radius_condition_before_sub}$ yields:
-\begin{equation*}
+\begin{equation}
+ \label{eqn:radius_condition}
\sage{problem1.radius_condition}
-\end{equation*}
+\end{equation}
\noindent
Expanding $\chern^{\beta_0}_2(u)$ in terms of $r$, $c$, $d$, and rearranging for
@@ -348,7 +349,7 @@ $d$ yields:
from plots_and_expressions import bgmlv3_d_upperbound_terms
\end{sagesilent}
-\begin{equation*}
+\begin{equation}
\label{eqn-bgmlv3_d_upperbound}
d \leq
\sage{bgmlv3_d_upperbound_terms.linear}
@@ -356,23 +357,19 @@ from plots_and_expressions import bgmlv3_d_upperbound_terms
\sage{bgmlv3_d_upperbound_terms.hyperbolic}
\qquad
\text{when }r>R
-\end{equation*}
+\end{equation}
\noindent
If $r=R$, then $\Delta(v-u)=(C-c)^2 \geq 0$ is always true, and for $r<R$ it gives a lower
bound on $d$, but it is weaker than the one given by the lower bound
-given by $\chern^P_2(u)>0$:
-\begin{equation*}
- d \geq
- \sage{bgmlv3_d_upperbound_terms.linear}
- + \sage{bgmlv3_d_upperbound_terms.const}
- + \sage{-bgmlv3_d_upperbound_terms.hyperbolic}
- \qquad
- \text{when }r<R
-\end{equation*}
+given by $\chern^P_2(u)>0$ if $u$ already satisfies Equations
+\ref{lem:eqn:cond-for-fixed-q}.
+We see this by comparing the unique terms from either bound:
+
{\color{red} THIS IS BECAUSE TODO}
+
\subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
\texorpdfstring{\ref{problem:problem-statement-2}}{2}}
\label{subsubsect:all-bounds-on-d-prob2}
--
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