From a47ee05d9a050ea23734726355d5c05a8e0fdc3c Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu, 15 Jun 2023 17:26:49 +0100
Subject: [PATCH] Enumerate suppositions in main lemma with letters
---
main.tex | 56 ++++++++++++--------------------------------------------
1 file changed, 12 insertions(+), 44 deletions(-)
diff --git a/main.tex b/main.tex
index 6c71811..d8567dc 100644
--- a/main.tex
+++ b/main.tex
@@ -253,7 +253,9 @@ Let $v$ and $u$ be Chern characters with positive ranks and $\Delta(v),
\noindent
Suppose that the following are satisfied:
-\begin{itemize}
+\bgroup
+\renewcommand{\labelenumi}{\alph{enumi}.}
+\begin{enumerate}
\item $u$ gives rise to a pseudo-wall for $v$, left of the vertical line $V_v$
\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)
@@ -262,7 +264,8 @@ Suppose that the following are satisfied:
$\nu_{\alpha,\beta}(\pm u)>\nu_{\alpha,\beta}(v)$ inside.
Where we use $+u$ or $-u$ depending on whether $(\beta,\alpha)$ is on the left
or right (resp.) of $V_u$.
-\end{itemize}
+\end{enumerate}
+\egroup
\noindent
Then we have the following:
@@ -284,17 +287,18 @@ $\Delta(u),\Delta(v) \geq 0$.
For the forwards implication, assume that the suppositions of the lemma are
satisfied. The pseudo-wall intersects $\Theta_v^-$, at some point $Q$ further up
-the hyperbola branch than $P$ (to satisfy second supposition). At $Q$, we have
+the hyperbola branch than $P$ (to satisfy supposition b). At $Q$, we have
$\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too. This means that $\Theta_u$ 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 the third supposition, only case 3 (green in
+of the base. However, considering supposition b, only case 3 (green in
figure) is valid. This is because we need $\nu_{P}(u)>0$ (or $\nu_{P}(-u)>0$ in
-case 1 involving $\Theta_u^+$). In passing, note that this implies consequence
-3. Recalling how the sign of $\nu_{\alpha,\beta}(\pm u)$ changes (illustrated in
+case 1 involving $\Theta_u^+$).
+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, note that this implies consequence 3.
\begin{sagesilent}
def hyperbola_intersection_plot():
@@ -415,48 +419,12 @@ implies that the assymptote for $\Theta_u^-$ is to the left of $\Theta_v^-$.
Consequence 3, 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
-$\alpha \to +\infty$. This implies suppositions 1 and 2, and that the
+$\alpha \to +\infty$. This implies suppositions a and b, and that the
characteristic curves for $u$ and $v$ must be in the configuration illustrated
in Fig \ref{fig:correct-hyperbol-intersection}. Recalling consequence 3 finally
-confirms supposition 3.
+confirms supposition c.
\end{proof}
-\begin{sagesilent}
-v = Chern_Char(3, 2, -2)
-u = Chern_Char(1, 0, 0)
-
-def charact_curve_with_wall_plot(u,v):
- alpha = stability.Tilt().alpha
- beta = stability.Tilt().beta
-
- coords_range = (beta, -5, 5), (alpha, 0, 5)
-
- charact_curve_plot = (
- implicit_plot(stability.Tilt().degree(u), *coords_range , rgbcolor = "red")
- + implicit_plot(stability.Tilt().degree(v), *coords_range )
- + line([(mu(v),0),(mu(v),5)], linestyle = "dashed", legend_label =
- r"$(3,2\ell,-4\ell^2/2)$")
- + line([(mu(u),0),(mu(u),5)], rgbcolor = "red", linestyle =
- "dashed", legend_label = r"$(1,0,0)$")
- + implicit_plot(stability.Tilt().wall_eqn(u,v)/alpha,
- *coords_range , rgbcolor = "black")
- )
- charact_curve_plot.xmax(1)
- charact_curve_plot.xmin(-2)
- charact_curve_plot.ymax(1.5)
- charact_curve_plot.axes_labels([r"$\beta$", r"$\alpha$"])
- return charact_curve_plot
-\end{sagesilent}
-
-\begin{figure}
- \centering
- \sageplot[width=\linewidth]{charact_curve_with_wall_plot(u,v)}
- \caption{}
- \label{fig:characteristic-curves-example}
-\end{figure}
-
-Talk about figure \ref{fig:characteristic-curves-example}.
-
\section{Loose Bounds on $\chern_0(E)$ for Semistabilizers Along Fixed
$\beta\in\QQ$}
--
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