From fde6e97902c60a5cb2a4fd9da3f10e725d60449c Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu, 20 Jul 2023 16:04:56 +0100
Subject: [PATCH] Add justificatiosn for problem 2
---
main.tex | 14 +++++++++-----
1 file changed, 9 insertions(+), 5 deletions(-)
diff --git a/main.tex b/main.tex
index edf57c8..f1c8d2e 100644
--- a/main.tex
+++ b/main.tex
@@ -560,8 +560,7 @@ are trying to solve for.
\begin{problem}[sufficiently large `left' pseudo-walls]
\label{problem:problem-statement-1}
-Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
-and $\beta_{-}(v) \in \QQ$.
+Fix a Chern character $v$ with positive rank, and $\Delta(v) \geq 0$.
The goal is to find all pseudo-semistabilizers $u$
which give circular pseudo-walls containing some fixed point
$P\in\Theta_v^-$.
@@ -598,13 +597,18 @@ $v-u$ for each solution $u$ of the problem.
Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
and $\beta_{-}(v) \in \QQ$.
-The goal is to find all solutions $u=(r,c\ell,d\ell^2)$
+The goal is to find all solutions $u$
to problem \ref{problem:problem-statement-1} with the choice
$P=(\beta_{-},0)$.
-
-This will give all circular pseudo-walls left of $V_v$.
\end{problem}
+This is a specialization of problem (\ref{problem:problem-statement-1})
+which will give all circular pseudo-walls left of $V_v$.
+This is because all circular walls left of $V_v$ intersect $\Theta_v^-$.
+The $\beta_{-}(v) \in \QQ$ condition is to ensure that there are finitely many
+solutions. As mentioned in the introduction (\ref{sec:intro}), this is known,
+however this will also be proved again in passing in this article.
+
\section{B.Schmidt's Solutions to the Problems}
--
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