Minor wording corrections

main.tex

@@ -238,15 +238,18 @@ Suppose that the following are satisfied:
 \item The pseudo-wall contains $p$ in it's interior
 	($P$ can be chosen to be the base of the left branch to target all left-walls)
 \item $u$ destabilizes $v$ going `inwards', that is,
-	$\nu_{\alpha,\beta}(u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
-	$\nu_{\alpha,\beta}(u)>\nu_{\alpha,\beta}(v)$ inside.
+	$\nu_{\alpha,\beta}(\pm u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
+	$\nu_{\alpha,\beta}(\pm u)>\nu_{\alpha,\beta}(v)$ inside.
+	Where we use $+u$ or $-u$ depending on whether $(\beta,\alpha)$ is on the left
+	or right (resp.) of the characteristic vertical line for $u$.
 \end{itemize}
 
 \noindent
 Then we have the following:
 \begin{itemize}
 	\item The pseudo-wall is left of $u$'s vertical characteristic line
-		(if this is a real wall then $v$ is being semistabilized by a positive rank object)
+		(if this is a real wall then $v$ is being semistabilized by an object with
+		Chern character $u$, not $-u$)
 	\item $\mu(u)<\mu(v)$, i.e., $u$'s vertical characteristic line is left of $v$'s vertical
 		characteristic line
 	\item $\chern_2^{P}(u)>0$
@@ -263,7 +266,7 @@ $\Delta(u),\Delta(v) \geq 0$.
 For the forwards implication, assume that the suppositions of the lemma are
 satisfied. The pseudo-wall intersects the characteristic hyperbola for $v$, at
 some point $Q$ further up the hyperbola branch than $P$ (to satisfy second
-supposition). At $Q$, we have $\mu_Q(v)=0$, and hence $\mu_Q(u)=0$ too.
+supposition). At $Q$, we have $\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too.
 This means that the characteristic hyperbola for $u$ must intersect that of $v$
 at $Q$. Considering the shapes of the hyperbolae alone, there are 3 distinct
 ways that they can intersect, as illustrated in Fig
@@ -273,11 +276,12 @@ $u$'s hyperbola, as well as the positions of the base.
 However, considering the third supposition, only case 3 (green in figure) is
 valid.
 This is because we need $\nu_{\alpha,\beta}(u)>0$
-($\nu_{\alpha,\beta}(-u)>0$ in case 1 involving the right hyperbola branch)
+(or $\nu_{\alpha,\beta}(-u)>0$ in case 1 involving the right hyperbola branch)
 for points $(\beta,\alpha)$ on $v$'s characteristic curve inside the pseudo-wall.
+In passing, note that this implies consequence 3.
 Recalling how the sign of $\nu_{\alpha,\beta}(\pm u)$ changes
 (illustrated in Fig \ref{fig:charact_curves_vis}), we can eliminate cases 1 and
-2. In passing, this implies consequence 3.
+2.
 
 \begin{sagesilent}
 def hyperbola_intersection_plot():