From 20c165a07d3c350dbe3923b84023fb2d9cd36a22 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu, 29 Jun 2023 22:06:16 +0100
Subject: [PATCH] Update statement of main lemma to be aware of quotient being
in tilt
---
main.tex | 28 +++++++++++-----------------
1 file changed, 11 insertions(+), 17 deletions(-)
diff --git a/main.tex b/main.tex
index 502ee04..55c7b68 100644
--- a/main.tex
+++ b/main.tex
@@ -284,36 +284,30 @@ the circular walls must be nested and non-intersecting.
\begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
\label{lem:pseudo_wall_numerical_tests}
-Let $v$ and $u$ be Chern characters with positive ranks and $\Delta(v),
-\Delta(u)\geq 0$. Let $P$ be a point on $\Theta_v^-$.
+Let $v$ and $u$ be Chern characters with $\Delta(v),
+\Delta(u)\geq 0$, and $v$ has positive rank. Let $P$ be a point on $\Theta_v^-$.
\noindent
-Suppose that the following are satisfied:
+The following conditions:
\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)
+\item $u$ is a pseudo-semistabilizer of $v$ at some point on $\Theta_v^-$ above
+ $P$
\item $u$ destabilizes $v$ going `inwards', that is,
- $\nu_{\alpha,\beta}(\pm u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
- $\nu_{\alpha,\beta}(\pm u)>\nu_{\alpha,\beta}(v)$ inside.
- Where we use $+u$ or $-u$ depending on whether $(\beta,\alpha)$ is on the left
- or right (resp.) of $V_u$.
+ $\nu_{\alpha,\beta}(u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
+ $\nu_{\alpha,\beta}(u)>\nu_{\alpha,\beta}(v)$ inside.
\end{enumerate}
\egroup
\noindent
-Then we have the following:
+are equivalent to the following more numerical conditions:
\begin{enumerate}
- \item The pseudo-wall is left of $V_u$
- (if this is a real wall then $v$ is being semistabilized by an object with
- Chern character $u$, not $-u$)
- \item $\beta(P)<\mu(u)<\mu(v)$, i.e., $V_u$ is strictly between $P$ and $V_v$.
+ \item $u$ has positive rank
+ \item $\beta(P)<\mu(u)<\mu(v)$, i.e. $V_u$ is strictly between $P$ and $V_v$.
+ \item $\chern_1^{\beta(P)}(v-u)\geq0$
\item $\chern_2^{P}(u)>0$
\end{enumerate}
-Furthermore, only the last two of these consequences are sufficient to recover
-all of the suppositions above.
\end{lemma}
\begin{proof}
--
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