From f12e230c3e84f82e0476bc9e9b8afc0d0d64f76d Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 12 Apr 2023 11:54:00 +0100
Subject: [PATCH] Write up to skeleton for hyperbola graph on twisted Cherns
---
main.tex | 17 ++++++++++++-----
1 file changed, 12 insertions(+), 5 deletions(-)
diff --git a/main.tex b/main.tex
index 3a6be6e..2b3b217 100644
--- a/main.tex
+++ b/main.tex
@@ -88,9 +88,10 @@ $\chern^\beta_1(E)$ is the imaginary component of the central charge
$\centralcharge_{\alpha,\beta}(E)$ and any element of $\firsttilt\beta$
satisfies $\chern^\beta_1 \geq 0$. This, along with additivity gives us, for any
destabilizing sequence [ref]:
-\[
+\begin{equation}
+ \label{eqn-tilt-cat-cond}
0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)
-\]
+\end{equation}
When finding Chern characters of potential destabilizers $E$ for some fixed
Chern character $\chern(F)$, this bounds $\chern_1(E)$.
@@ -98,9 +99,15 @@ Chern character $\chern(F)$, this bounds $\chern_1(E)$.
The Bogomolov form applied to the twisted Chern character is the same as the
normal one. So $0 \leq \Delta(E)$ yields:
-\[
- \chern^\beta_0 \chern^\beta_2 \leq \left(\chern^\beta_1\right)^2
-\]
+\begin{equation}
+ \label{eqn-bgmlv-on-E}
+ \chern^\beta_0(E) \chern^\beta_2(E) \leq \left(\chern^\beta_1(E)\right)^2
+\end{equation}
+
+The restrictions on $\chern^\beta_0(E)$ and $\chern^\beta_2(E)$
+is best seen with the following graph:
+
+% TODO: hyperbola restriction graph (shaded)
\section{Section 3}
--
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