From 8ba0438a0fe2941a7d96680c77c36a8b38de6e4b Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 13 Sep 2023 22:53:50 +0100
Subject: [PATCH] Extend numerical formulations to rank 0
---
main.tex | 18 ++++++++++++------
1 file changed, 12 insertions(+), 6 deletions(-)
diff --git a/main.tex b/main.tex
index fcfcac8..504b0fe 100644
--- a/main.tex
+++ b/main.tex
@@ -506,7 +506,8 @@ 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. Let $P$ be a point on $\Theta_v^-$.
+\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:
@@ -635,7 +636,8 @@ are trying to solve for.
\begin{problem}[sufficiently large `left' pseudo-walls]
\label{problem:problem-statement-1}
-Fix a Chern character $v$ with non-negative rank, and $\Delta(v) \geq 0$.
+Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
+and $\Delta(v) \geq 0$.
The goal is to find all pseudo-semistabilizers $u$
which give circular pseudo-walls containing some fixed point
$P\in\Theta_v^-$.
@@ -670,8 +672,8 @@ $v-u$ for each solution $u$ of the problem.
\begin{problem}[all `left' pseudo-walls]
\label{problem:problem-statement-2}
-Fix a Chern character $v$ with non-negative rank, $\Delta(v) \geq 0$,
-and $\beta_{-}(v) \in \QQ$.
+Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
+$\Delta(v) \geq 0$, 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}
@@ -697,7 +699,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
\begin{lemma}[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
\label{lem:num_test_prob1}
- Given a Chern character $v$ with positive rank and $\Delta(v) \geq 0$,
+ Given a Chern character $v$ with non-negative rank
+ (with $\chern_1(v)>0$ if rank 0)
+ and $\Delta(v) \geq 0$,
and a choice of point $P$ on $\Theta_v^-$.
Solutions $u=(r,c\ell,\frac{e}{\lcm(m,2)}\ell^2)$
to problem \ref{problem:problem-statement-1}.
@@ -718,7 +722,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
\end{lemma}
\begin{proof}
- Consider the context of $v$ being a Chern character with positive rank and
+ Consider the context of $v$ being a Chern character with non-negative rank
+ (and $\chern_1(v)>0$ if rank 0)
+ and
$\Delta \geq 0$, and $u$ being a Chern character with $\Delta(u) \geq 0$.
Lemma \ref{lem:pseudo_wall_numerical_tests} gives that the remaining
conditions for $u$ being a solution to problem
--
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