Complete proof of numerical condition equivalence in rank 0 case

tex/setting-and-problems.tex

@@ -162,11 +162,10 @@ are equivalent to the following more numerical conditions:
 \end{enumerate}
 \end{lemma}
 
-\begin{proof}
+\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.
 
-
 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 $v$.
@@ -244,10 +243,47 @@ Therefore, it's also a pseudo-semistabiliser further along the circle at $Q$
 (supposition a).
 Finally, consequence 4 along with $P$ being to the left of $V_u$ implies
 $\nu_P(u) > 0$ giving supposition b.
+\end{proof}
+\begin{sagesilent}
+from rank_zero_case_curves import pseudo_semistab_char_curves_rank_zero
+\end{sagesilent}
+\begin{figure}
+	\centering
+	\sageplot[width=\textwidth]{pseudo_semistab_char_curves_rank_zero}
+	\caption{characteristic curves configuration for pseudo-semistabiliser of $v$
+	with rank 0 destabilising `downwards'}
+	\label{fig:hyperbol-intersection-rank-zero}
+\end{figure}
+\begin{proof}[Proof for $\chern_0(v)=0$ case.]
+Let $u,v$ be Chern characters with
+$\Delta(u),\Delta(v) \geq 0$, and $\chern_0(v)=0$ but $\chern_1(v)>0$.
+So $\Theta_v^-$ is the vertical line at $\beta = \beta_{-}(v)$, and
+$\mu(v) = +\infty$.
+
+For the forward implication, assume suppositions a and b hold.
+Let $Q$ be the point above $P$ on $\Theta_v^-$ where $u$ is a
+pseudo-semistabiliser of $v$. So $\Theta_u$ intersects $\Theta_v^-$ at $Q$.
+Suppose, seeking a contradiction that $\chern_0(u)=0$,
+then $\Theta_u = \Theta_u^-$ is also a vertical line,
+and must then be the same line as $\Theta_v^-$ to intersect $Q$.
+This would imply $\QQ u = \QQ v$ and then $u$ would not destabilise $v$ at any
+stability condition.
+So we must have either $\chern_0(u) < 0$, in which case $\Theta_u^+$ is the
+branch of $\Theta_u$ going through $Q$; or $\chern_0(u) > 0$, in which it is
+$\Theta_u^-$ instead.
+The latter case is the only one which could satisfy supposition b, about $u$
+destabilising $v$ going `down' $\Theta_v^-$.
+Which then forces the configuration of characteristic curves shown in Figure
+\ref{fig:hyperbol-intersection-rank-zero}.
+The positions of the characteristic curves ensures the numerical conditions 1, 2
+and 5. The other conditions follow from the definition of $u$ being a
+pseudo-semistabiliser of $v$.
 
-The case with rank 0 can be handled the same way.
-% TODO expand this case too
-
+Conversely, suppose that the numerical conditions 1-5 are satisfied,
+then this forces the configuration of characteristic curves shown in Figure
+\ref{fig:hyperbol-intersection-rank-zero}.
+This ensures that $u$ is a pseudo-semistabiliser of $v$ at $u$ destabilising
+going down $\Theta_v^-$.
 \end{proof}
 
 \begin{remark}
@@ -295,7 +331,7 @@ The case with rank 0 can be handled the same way.
 
 \section{The Problem: Finding Pseudo-walls}
 
-As hinted in the introduction (\ref{sec:intro}), the main motivation of the
+As hinted in the introduction to this Part \ref{part:fin-walls}, the main motivation of the
 results in this article are not only the bounds on pseudo-semistabiliser
 ranks;
 but also applications for finding a list (comprehensive or subset) of