Define beta +- in zero/positive rank cases

main.tex

@@ -356,11 +356,27 @@ degenerate_characteristic_curves
 \label{fig:charact_curves_vis}
 \end{figure}
 
+\begin{definition}[$\beta_{\pm}$]
+	\label{dfn:beta_pm}
+	Given a formal Chern character $v$ with positive rank, we define $\beta_{\pm}(v)$ to be
+	the $\beta$-coordinate of where $\Theta_v^{\pm}$ meets the $\beta$-axis:
+	\[
+		\beta_\pm(R,C\ell,D\ell^2) = \frac{C \pm \sqrt{C^2-2RD}}{R}
+	\]
+	\noindent
+	In particular, this means $\beta_\pm(v)$ are the two roots of the quadratic
+	equation $\chern_2^{\beta}(v)=0$.
+
+	This definition will be extended to the rank 0 case in definition \ref{dfn:beta_-_rank0}.
+\end{definition}
+
+
+\subsubsection{Rank Zero Case}
+
 \begin{sagesilent}
 from rank_zero_case import Theta_v_plot
 \end{sagesilent}
 
-\subsubsection{Rank Zero Case}
 \begin{fact}[Geometry of Characteristic Curves in Rank 0 Case]
 The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
 as well as the restrictions on $v$, when $\chern_0(v)=0$ and $\chern_1(v)>0$:
@@ -387,7 +403,7 @@ Indeed:
 \begin{align*}
 	\mu\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C}{\varepsilon} &\longrightarrow +\infty
 	\\
-	\text{as} \: 0<\varepsilon &\longrightarrow 0
+	\text{as} \:\: 0<\varepsilon &\longrightarrow 0
 \end{align*}
 So we can view $V_v$ as moving off infinitely to the right, with $\Theta_v^+$ even further.
 But also, considering the base point of $\Theta_v^-$:
@@ -404,6 +420,19 @@ For this reason, I will refer to the whole of $\Theta_v$ in the rank zero case
 as $\Theta_v^-$ to be able to use the same terminology in both positive rank
 and rank zero cases.
 
+\begin{definition}[Extending $\beta_-$ to rank 0 case]
+	\label{dfn:beta_-_rank0}
+	Given a formal Chern character $v$ with rank 0 and $\chern_1(v)>0$, we define
+	$\beta_-(v)$ to be the $\beta$-coordinate of point where $\Theta_v$ meets the
+	$\beta$-axis:
+	\[
+		\beta_-(0,C\ell,D\ell^2) = \frac{D}{C}
+	\]
+	\noindent
+	If $\beta_+$ were also to be generalised to the rank 0 case, we would consider
+	its value to be $+\infty$ due to the discussion above.
+\end{definition}
+
 
 \subsection{Relevance of $V_v$}
 \label{subsect:relevance-of-V_v}