Adjust tighter bounds section to more general surface

main.tex

@@ -838,7 +838,6 @@ As opposed to only eliminating possible values of $\chern_0(E)$ for which all
 corresponding $\chern_1^{\beta}(E)$ fail one of the inequalities (which is what
 was implicitly happening before).
 
-% NOTE FUTURE: surface specialization
 
 First, let us fix a Chern character for $F$, and some pseudo-semistabilizer
 $u$ which is a solution to problem
@@ -850,10 +849,10 @@ Take $\beta = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
 
 \begin{align}
 	\chern(F) =\vcentcolon\: v \:=& \:(R,C\ell,D\ell^2)
-		&& \text{where $R,C,2D\in \ZZ$}
+	&& \text{where $R,C\in \ZZ$ and $D\in \frac{1}{\lcm(m,2)}\ZZ$}
 	\\
 	u \coloneqq& \:(r,c\ell,d\ell^2)
-		&& \text{where $r,c,2d\in \ZZ$}
+	&& \text{where $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$}
 \end{align}
  
 
@@ -1120,10 +1119,10 @@ for the bounds on $d$ in terms of $r$ is illustrated in figure
 (\ref{fig:d_bounds_xmpl_gnrc_q}).
 The question of whether there are pseudo-destabilizers of arbitrarily large
 rank, in the context of the graph, comes down to whether there are points
-$(r,d) \in \ZZ \oplus \frac{1}{2} \ZZ$ (with large $r$)
+$(r,d) \in \ZZ \oplus \frac{1}{\lcm(m,2)} \ZZ$ (with large $r$)
 % TODO have a proper definition for pseudo-destabilizers/walls
 that fit above the yellow line (ensuring positive radius of wall) but below the
-blue and green (ensuring $\Delta(E), \Delta(G) > 0$).
+blue and green (ensuring $\Delta(u), \Delta(v-u) > 0$).
 These lines have the same assymptote at $r \to \infty$
 (eqns \ref{eqn:bgmlv2_d_bound_betamin},
 \ref{eqn:bgmlv3_d_bound_betamin},