Relate the Tigter bounds section to problems 1,2

main.tex

@@ -960,6 +960,10 @@ First, let us fix a Chern character for $F$, and some pseudo-semistabilizer
 $u$ which is a solution to problem
 \ref{problem:problem-statement-1} or
 \ref{problem:problem-statement-2}.
+Take $\beta = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
+\ref{problem:problem-statement-1}
+(or $\beta = \beta_{-}$ for problem \ref{problem:problem-statement-2}).
+
 \begin{align}
 	\chern(F) =\vcentcolon\: v \:=& \:(R,C\ell,D\ell^2)
 		&& \text{where $R,C,2D\in \ZZ$}
@@ -1006,7 +1010,8 @@ c_in_terms_of_q = c_lower_bound + q
 \end{equation}
 
 Furthermore, $\chern_1 \in \ZZ$ so we only need to consider
-$q \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta}(F)]$.
+$q \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta}(F)]$,
+where $n$ is the denominator of $\beta$.
 For the next subsections, we consider $q$ to be fixed with one of these values,
 and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.