From 495b5ec9b0e3a34a30c8842c9a5cbec30ea5738c Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 2 Jul 2024 16:14:33 +0100
Subject: [PATCH] Phrase a proof in terms of the problem
---
tex/bounds-on-semistabilisers.tex | 8 ++++----
1 file changed, 4 insertions(+), 4 deletions(-)
diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index 1451d05..372e135 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -37,13 +37,13 @@ bounded above by:
\begin{proof}
The Bogomolov form applied to the twisted Chern character is the same as the
-untwisted one. So $0 \leq \Delta(E)$
+untwisted one. So $0 \leq \Delta(u)$
(condition 2 from Corollary \ref{cor:num_test_prob2})
yields:
\begin{equation}
\label{eqn-bgmlv-on-E}
- 2\chern^\beta_0(E) \chern^\beta_2(E) \leq \chern^\beta_1(E)^2
+ 2\chern^\beta_0(u) \chern^\beta_2(u) \leq \chern^\beta_1(u)^2
\end{equation}
\noindent
@@ -52,12 +52,12 @@ condition 5 from Corollary \ref{cor:num_test_prob2}
gives:
\begin{equation}
\label{eqn-tilt-cat-cond}
- 0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)
+ 0 < \chern^\beta_1(u) < \chern^\beta_1(v)
\end{equation}
% FUTURE maybe ref this back to some definition of first tilt
\noindent
-The restrictions on $\chern^{\beta_-}_0(E)$ and $\chern^{\beta_-}_2(E)$
+The restrictions on $\chern^{\beta_-}_0(u)$ and $\chern^{\beta_-}_2(v)$
is best seen with the following graph:
% TODO: hyperbola restriction graph (shaded)
--
GitLab