Sort out section about bounds on r in problem 1

tex/bounds-on-semistabilisers.tex

@@ -229,6 +229,7 @@ was implicitly happening before in the proof of Theorem
 \end{proof}
 
 \subsection{Bounds on \texorpdfstring{`$d$'}{d}-values for Solutions of Problems}
+\label{subsec:bounds-on-d}
 
 Let $v$ be a Chern character with $\Delta(v)\geq 0$,
 $\chern_0(v)>0$, or $\chern_0(v)=0$ and $\chern_1(v)>0$,
@@ -603,7 +604,7 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
 
 \begin{theorem}[Problem \ref{problem:problem-statement-1} upper Bound on $r$]
 \label{lem:prob1:r_bound}
-	Let $u$ be a solution to problem \ref{problem:problem-statement-1}
+	Let $u$ be a solution to Problem \ref{problem:problem-statement-1}
 	and $q\coloneqq\chern_1^{B}(u)$.
 	Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression:
 	\begin{equation}
@@ -612,9 +613,15 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
 \end{theorem}
 
 \begin{proof}
-	Recall that $d\coloneqq\chern_2(u)$ has two upper bounds in terms of $r$: in
+	Lemma \ref{lem:fixed-q-semistabs-criterion} gives us that any solution $u$
+	must be of the form in Equation \ref{eqn:u-coords} and
+	satisfy Equations \ref{lem:eqn:cond-for-fixed-q} as well as the three
+	conditions $\chern^P_2(u)>0$, $\Delta(u) \geq 0$, and $\Delta(v-u) \geq 0$.
+	Subsection \ref{subsec:bounds-on-d} equates these latter three conditions
+	(provided Equations \ref{lem:eqn:cond-for-fixed-q})
+	to upper bounds on $d$ given by
 	Equations \ref{eqn:prob1:bgmlv2} and \ref{eqn:prob1:bgmlv3};
-	and one lower bound: in Equation \ref{eqn:prob1:radiuscond}.
+	and one lower bound given by Equation \ref{eqn:prob1:radiuscond}.
 
 	Solving for the lower bound in Equation \ref{eqn:prob1:radiuscond} being
 	less than the upper bound in Equation \ref{eqn:prob1:bgmlv2} yields:
@@ -622,18 +629,20 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
 	r<\sage{problem1.positive_intersection_bgmlv2}
 	\end{equation}
 
+	\noindent
 	Similarly, but with the upper bound in Equation \ref{eqn:prob1:bgmlv3}, gives:
 	\begin{equation}
 	r<\sage{problem1.positive_intersection_bgmlv3}
 	\end{equation}
 
+	\noindent
 	Therefore, $r$ is bounded above by the minimum of these two expressions which
 	can then be factored into the expression given in the Lemma.
 	
 \end{proof}
 
 The above Lemma \ref{lem:prob1:r_bound} gives an upper bound on $r$ in terms of $q$.
-But given that $0 \leq q \leq \chern_1^{B}(v)$, we can take the maximum of this
+But given that $0 < q < \chern_1^{\beta_0}(v)$, we can take the maximum of this
 bound, over $q$ in this range, to get a simpler (but weaker) bound in the
 following Lemma \ref{lem:prob1:convenient_r_bound}.