Complete one outstanding TODO

tex/setting-and-problems.tex

@@ -157,12 +157,11 @@ are equivalent to the following more numerical conditions:
 	\item $\beta(P)<\mu(u)<\mu(v)$, i.e. $V_u$ is strictly between $P$ and $V_v$.
 		\label{lem:ps-wall-num-test:num-cond-slope}
 	\item $\chern_1^{\beta(P)}(u) < \chern_1^{\beta(P)}(v)$
-	\item	$\Delta(v-u) \geq 0$
+	\item $\Delta(v-u) \geq 0$
 	\item $\chern_2^{P}(u)>0$
 \end{enumerate}
 \end{lemma}
 
-\noindent\textbf{\color{red} TODO ADJUST TO SLIGHTLY STRONGER CONDITION 3}
 \begin{proof}[Proof for $\chern_0(v)>0$ case.]
 Let $u,v$ be Chern characters with
 $\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
@@ -170,9 +169,23 @@ $\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
 For the forwards implication, assume that the suppositions of the Lemma are
 satisfied. Let $Q$ be the point on $\Theta_v^-$ (above $P$) where $u$ is a
 pseudo-semistabiliser of $v$.
-Firstly, consequence 3 is part of the definition for $u$ being a
+Firstly, consequence 4 is part of the definition for $u$ being a
 pseudo-semistabiliser at a point with same $\beta$ value of $P$ (since the
 pseudo-wall surrounds $P$).
+Also, $u$ is a pseudo-semistabiliser along the pseudo-wall
+which surround $P$, so by definition of pseudo-semistabilisers,
+${
+    0 \leq \chern_1^\beta(u) \leq \chern_1^\beta(v)
+}$
+for all $(\alpha, \beta)$ on the pseudo-wall.
+In particular, this holds for $\beta$ in a neighbourhood of $\beta(P)$,
+so we can conclude
+${
+    0 < \chern_1^{\beta(P)}(u) < \chern_1^{\beta(P)}(v)
+}$
+(consequence 5).
+Notice that $0 < \chern_1^{\beta(P)}(u)$
+follows from consequences 1 and 2, this is why it is not included in consequence 5.
 If $u$ were to have 0 rank, it's tilt slope would be decreasing as $\beta$
 increases, contradicting supposition b. So $u$ must have strictly non-zero rank,
 and we can consider it's characteristic curves (or that of $-u$ in case of
@@ -218,8 +231,8 @@ have positive rank (consequence 1)
 to ensure that $\chern_1^{\beta(P)} \geq 0$ (since the pseudo-wall passed over
 $P$).
 Furthermore, $P$ being on the left of $V_u$ implies
-$\chern_1^{\beta{P}}(u) \geq 0$,
-and therefore $\chern_2^{P}(u) > 0$ (consequence 4) to satisfy supposition b.
+$\chern_1^{\beta(P)}(u) > 0$,
+and therefore $\chern_2^{P}(u) > 0$ (consequence 5) to satisfy supposition b.
 Next considering the way the hyperbolae intersect, we must have $\Theta_u^-$ taking a
 base-point to the right $\Theta_v$, but then, further up, crossing over to the
 left side. The latter fact implies that the assymptote for $\Theta_u^-$ must be
@@ -229,9 +242,9 @@ must have $\mu(u)<\mu(v)$ (second part of consequence 2),
 that is, $V_u$ is strictly to the left of $V_v$.
 
 
-Conversely, suppose that the consequences 1-4 are satisfied. Consequence 2
+Conversely, suppose that the consequences 1-5 are satisfied. Consequence 2
 implies that the assymptote for $\Theta_u^-$ is to the left of $\Theta_v^-$.
-Consequence 4, along with $\beta(P)<\mu(u)$, implies that $P$ must be in the
+Consequences 4 and 5, along with $\beta(P)<\mu(u)$, implies that $P$ must be in the
 region left of $\Theta_u^-$. These two facts imply that $\Theta_u^-$ is to the
 right of $\Theta_v^-$ at $\alpha=\alpha(P)$, but crosses to the left side as
 $\alpha \to +\infty$, intersection at some point $Q$ above $P$.