Write up to skeleton for hyperbola graph on twisted Cherns

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}