From a500675b10d250a24ed64c48050071d510c405e4 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 13 Sep 2023 22:29:47 +0100
Subject: [PATCH] Extend problem statement to rank 0
---
main.tex | 13 ++++++++++---
1 file changed, 10 insertions(+), 3 deletions(-)
diff --git a/main.tex b/main.tex
index 8550862..fcfcac8 100644
--- a/main.tex
+++ b/main.tex
@@ -372,6 +372,7 @@ degenerate_characteristic_curves
\subsubsection{Rank Zero Case}
+\label{subsubsect:rank-zero-case-charact-curves}
\begin{sagesilent}
from rank_zero_case import Theta_v_plot
@@ -634,7 +635,7 @@ 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, and $\Delta(v) \geq 0$.
+Fix a Chern character $v$ with non-negative rank, and $\Delta(v) \geq 0$.
The goal is to find all pseudo-semistabilizers $u$
which give circular pseudo-walls containing some fixed point
$P\in\Theta_v^-$.
@@ -669,19 +670,25 @@ $v-u$ for each solution $u$ of the problem.
\begin{problem}[all `left' pseudo-walls]
\label{problem:problem-statement-2}
-Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
+Fix a Chern character $v$ with non-negative rank, $\Delta(v) \geq 0$,
and $\beta_{-}(v) \in \QQ$.
The goal is to find all pseudo-semistabilizers $u$ which give circular
pseudo-walls on the left side of $V_v$.
\end{problem}
This is a specialization of problem (\ref{problem:problem-statement-1})
-with the choice $P=(\beta_{-},0)$.
+with the choice $P=(\beta_{-},0)$, the point where $\Theta_v^-$ meets the
+$\beta$-axis.
This is because all circular walls left of $V_v$ intersect $\Theta_v^-$ (once).
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.
+This description still holds for the case of rank 0 case if we consider $V_v$ to
+be infinitely far to the right
+(see section \ref{subsubsect:rank-zero-case-charact-curves}).
+Note also that the $\beta_-(v)$ condition always holds for $v$ rank 0.
+
\subsection{Numerical Formulations of the Problems}
The problems introduced in this section are phrased in the context of stability
--
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