From 7e8de853f006cb42572eb076788ee691f1e44094 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Mon, 12 Jun 2023 17:55:50 +0100
Subject: [PATCH] Minor wording corrections
---
main.tex | 16 ++++++++++------
1 file changed, 10 insertions(+), 6 deletions(-)
diff --git a/main.tex b/main.tex
index 99421a1..6596ef5 100644
--- a/main.tex
+++ b/main.tex
@@ -238,15 +238,18 @@ Suppose that the following are satisfied:
\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 $u$ destabilizes $v$ going `inwards', that is,
- $\nu_{\alpha,\beta}(u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
- $\nu_{\alpha,\beta}(u)>\nu_{\alpha,\beta}(v)$ inside.
+ $\nu_{\alpha,\beta}(\pm u)<\nu_{\alpha,\beta}(v)$ 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 the characteristic vertical line for $u$.
\end{itemize}
\noindent
Then we have the following:
\begin{itemize}
\item The pseudo-wall is left of $u$'s vertical characteristic line
- (if this is a real wall then $v$ is being semistabilized by a positive rank object)
+ (if this is a real wall then $v$ is being semistabilized by an object with
+ Chern character $u$, not $-u$)
\item $\mu(u)<\mu(v)$, i.e., $u$'s vertical characteristic line is left of $v$'s vertical
characteristic line
\item $\chern_2^{P}(u)>0$
@@ -263,7 +266,7 @@ $\Delta(u),\Delta(v) \geq 0$.
For the forwards implication, assume that the suppositions of the lemma are
satisfied. The pseudo-wall intersects the characteristic hyperbola for $v$, at
some point $Q$ further up the hyperbola branch than $P$ (to satisfy second
-supposition). At $Q$, we have $\mu_Q(v)=0$, and hence $\mu_Q(u)=0$ too.
+supposition). At $Q$, we have $\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too.
This means that the characteristic hyperbola for $u$ must intersect that of $v$
at $Q$. Considering the shapes of the hyperbolae alone, there are 3 distinct
ways that they can intersect, as illustrated in Fig
@@ -273,11 +276,12 @@ $u$'s hyperbola, as well as the positions of the base.
However, considering the third supposition, only case 3 (green in figure) is
valid.
This is because we need $\nu_{\alpha,\beta}(u)>0$
-($\nu_{\alpha,\beta}(-u)>0$ in case 1 involving the right hyperbola branch)
+(or $\nu_{\alpha,\beta}(-u)>0$ in case 1 involving the right hyperbola branch)
for points $(\beta,\alpha)$ on $v$'s characteristic curve inside the pseudo-wall.
+In passing, note that this implies consequence 3.
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, this implies consequence 3.
+2.
\begin{sagesilent}
def hyperbola_intersection_plot():
--
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