Start thinking about gap for 3rd condition when r<R

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}