From 3595122db902d676be3d14663eddf99231c0ab3e Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu, 20 Jun 2024 16:10:10 +0100
Subject: [PATCH] Move pseudo semistab charact curve material to main content

---
 tex/characteristic-curves.tex | 128 ----------------------------------
 tex/content.tex               | 128 ++++++++++++++++++++++++++++++++++
 2 files changed, 128 insertions(+), 128 deletions(-)

diff --git a/tex/characteristic-curves.tex b/tex/characteristic-curves.tex
index c3ba84e..e3a9a00 100644
--- a/tex/characteristic-curves.tex
+++ b/tex/characteristic-curves.tex
@@ -212,131 +212,3 @@ This fact, along with the hindsight knowledge that non-vertical walls are
 circles with centers on the $\beta$-axis, gives an alternative view to see that
 the circular walls must be nested and non-intersecting.
 
-\subsection{Characteristic Curves for Pseudo-semistabilizers}
-
-These characteristic curves introduced are convenient tools to think about the
-numerical conditions that can be used to test for pseudo-semistabilizers, and
-for solutions to the problems
-(\ref{problem:problem-statement-1},\ref{problem:problem-statement-2})
-tackled in this article (to be introduced later).
-In particular, problem (\ref{problem:problem-statement-1}) will be translated to
-a list of numerical inequalities on it's solutions $u$.
-% ref to appropriate Lemma when it's written
-
-The next Lemma is a key to making this translation and revolves around the
-geometry and configuration of the characteristic curves involved in a
-semistabilizing sequence.
-
-\begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
-\label{lem:pseudo_wall_numerical_tests}
-Let $v$ and $u$ be Chern characters with $\Delta(v),
-\Delta(u)\geq 0$, and $v$ has non-negative rank (and $\chern_1(v)>0$ if rank 0).
-Let $P$ be a point on $\Theta_v^-$.
-
-\noindent
-The following conditions:
-\begin{enumerate}
-\item $u$ is a pseudo-semistabilizer of $v$ at some point on $\Theta_v^-$ above
-	$P$
-\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.
-\end{enumerate}
-
-\noindent
-are equivalent to the following more numerical conditions:
-\begin{enumerate}
-	\item $u$ has positive rank
-	\item $\beta(P)<\mu(u)<\mu(v)$, i.e. $V_u$ is strictly between $P$ and $V_v$.
-	\item $\chern_1^{\beta(P)}(u) \leq \chern_1^{\beta(P)}(v)$, $\Delta(v-u) \geq 0$
-	\item $\chern_2^{P}(u)>0$
-\end{enumerate}
-\end{lemma}
-
-\begin{proof}
-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-semistabilizer of $v$.
-Firstly, consequence 3 is part of the definition for $u$ being a
-pseudo-semistabilizer at a point with same $\beta$ value of $P$ (since the
-pseudo-wall surrounds $P$).
-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
-negative rank).
-$\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too. This means that $\Theta_{\pm u}$ must
-intersect $\Theta_v^-$ at $Q$. Considering the shapes of the hyperbolae alone,
-there are 3 distinct ways that they can intersect, as illustrated in Fig
-\ref{fig:hyperbol-intersection}. These cases are distinguished by whether it is
-the left, or the right branch of $\Theta_u$ involved, as well as the positions
-of the base. However, considering supposition b, only case 3 (green in
-figure) is possible. This is because we need $\nu_{P}(u)>0$ (or $\nu_{P}(-u)>0$ in
-case 1 involving $\Theta_u^+$), to satisfy supposition b.
-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.
-
-\begin{sagesilent}
-from characteristic_curves import \
-hyperbola_intersection_plot, \
-correct_hyperbola_intersection_plot
-\end{sagesilent}
-
-\begin{figure}
-\begin{subfigure}[t]{0.48\textwidth}
-	\centering
-	\sageplot[width=\textwidth]{hyperbola_intersection_plot()}
-	\caption{Three ways the characteristic hyperbola for $u$ can intersect the left
-	branch of the characteristic hyperbola for $v$}
-	\label{fig:hyperbol-intersection}
-\end{subfigure}
-\hfill
-\begin{subfigure}[t]{0.48\textwidth}
-	\centering
-	\sageplot[width=\textwidth]{correct_hyperbola_intersection_plot()}
-	\caption{Closer look at characteristic curves for valid case}
-	\label{fig:correct-hyperbol-intersection}
-\end{subfigure}
-\end{figure}
-
-Fixing attention on the only possible case (2), illustrated in Fig
-\ref{fig:correct-hyperbol-intersection}.
-$P$ is on the left of $V_{\pm u}$ (first part of consequence 2), so $u$ must
-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.
-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
-to the left of the one for $\Theta_v^-$. Given that they are parallel and
-intersect the $\beta$-axis at $\beta=\mu(u)$ and $\beta=\mu(v)$ respectively. We
-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
-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
-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 $P$.
-This implies that the characteristic curves for $u$ and $v$ are in the
-configuration illustrated in Fig \ref{fig:correct-hyperbol-intersection}.
-We then have $\nu(u)=\nu(v)$ along a circle to the left of $V_u$ reaching it's
-apex at $Q$, and encircling $P$. This along with consequence 3 implies that $u$
-is a pseudo-semistabilizer at the point on the circle with $\beta=\beta(P)$.
-Therefore, it's also a pseudo-semistabilizer 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.
-
-The case with rank 0 can be handled the same way.
-
-\end{proof}
-
diff --git a/tex/content.tex b/tex/content.tex
index df56379..153cb8f 100644
--- a/tex/content.tex
+++ b/tex/content.tex
@@ -1,3 +1,131 @@
+\subsection{Characteristic Curves for Pseudo-semistabilizers}
+
+These characteristic curves introduced are convenient tools to think about the
+numerical conditions that can be used to test for pseudo-semistabilizers, and
+for solutions to the problems
+(\ref{problem:problem-statement-1},\ref{problem:problem-statement-2})
+tackled in this article (to be introduced later).
+In particular, problem (\ref{problem:problem-statement-1}) will be translated to
+a list of numerical inequalities on it's solutions $u$.
+% ref to appropriate Lemma when it's written
+
+The next Lemma is a key to making this translation and revolves around the
+geometry and configuration of the characteristic curves involved in a
+semistabilizing sequence.
+
+\begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
+\label{lem:pseudo_wall_numerical_tests}
+Let $v$ and $u$ be Chern characters with $\Delta(v),
+\Delta(u)\geq 0$, and $v$ has non-negative rank (and $\chern_1(v)>0$ if rank 0).
+Let $P$ be a point on $\Theta_v^-$.
+
+\noindent
+The following conditions:
+\begin{enumerate}
+\item $u$ is a pseudo-semistabilizer of $v$ at some point on $\Theta_v^-$ above
+	$P$
+\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.
+\end{enumerate}
+
+\noindent
+are equivalent to the following more numerical conditions:
+\begin{enumerate}
+	\item $u$ has positive rank
+	\item $\beta(P)<\mu(u)<\mu(v)$, i.e. $V_u$ is strictly between $P$ and $V_v$.
+	\item $\chern_1^{\beta(P)}(u) \leq \chern_1^{\beta(P)}(v)$, $\Delta(v-u) \geq 0$
+	\item $\chern_2^{P}(u)>0$
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+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-semistabilizer of $v$.
+Firstly, consequence 3 is part of the definition for $u$ being a
+pseudo-semistabilizer at a point with same $\beta$ value of $P$ (since the
+pseudo-wall surrounds $P$).
+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
+negative rank).
+$\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too. This means that $\Theta_{\pm u}$ must
+intersect $\Theta_v^-$ at $Q$. Considering the shapes of the hyperbolae alone,
+there are 3 distinct ways that they can intersect, as illustrated in Fig
+\ref{fig:hyperbol-intersection}. These cases are distinguished by whether it is
+the left, or the right branch of $\Theta_u$ involved, as well as the positions
+of the base. However, considering supposition b, only case 3 (green in
+figure) is possible. This is because we need $\nu_{P}(u)>0$ (or $\nu_{P}(-u)>0$ in
+case 1 involving $\Theta_u^+$), to satisfy supposition b.
+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.
+
+\begin{sagesilent}
+from characteristic_curves import \
+hyperbola_intersection_plot, \
+correct_hyperbola_intersection_plot
+\end{sagesilent}
+
+\begin{figure}
+\begin{subfigure}[t]{0.48\textwidth}
+	\centering
+	\sageplot[width=\textwidth]{hyperbola_intersection_plot()}
+	\caption{Three ways the characteristic hyperbola for $u$ can intersect the left
+	branch of the characteristic hyperbola for $v$}
+	\label{fig:hyperbol-intersection}
+\end{subfigure}
+\hfill
+\begin{subfigure}[t]{0.48\textwidth}
+	\centering
+	\sageplot[width=\textwidth]{correct_hyperbola_intersection_plot()}
+	\caption{Closer look at characteristic curves for valid case}
+	\label{fig:correct-hyperbol-intersection}
+\end{subfigure}
+\end{figure}
+
+Fixing attention on the only possible case (2), illustrated in Fig
+\ref{fig:correct-hyperbol-intersection}.
+$P$ is on the left of $V_{\pm u}$ (first part of consequence 2), so $u$ must
+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.
+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
+to the left of the one for $\Theta_v^-$. Given that they are parallel and
+intersect the $\beta$-axis at $\beta=\mu(u)$ and $\beta=\mu(v)$ respectively. We
+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
+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
+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 $P$.
+This implies that the characteristic curves for $u$ and $v$ are in the
+configuration illustrated in Fig \ref{fig:correct-hyperbol-intersection}.
+We then have $\nu(u)=\nu(v)$ along a circle to the left of $V_u$ reaching it's
+apex at $Q$, and encircling $P$. This along with consequence 3 implies that $u$
+is a pseudo-semistabilizer at the point on the circle with $\beta=\beta(P)$.
+Therefore, it's also a pseudo-semistabilizer 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.
+
+The case with rank 0 can be handled the same way.
+
+\end{proof}
+
 \section{The Problem: Finding Pseudo-walls}
 
 As hinted in the introduction (\ref{sec:intro}), the main motivation of the
-- 
GitLab