Introduce rank 0 case up to main lemma

main.tex

@@ -289,8 +289,8 @@ Given a Chern character $v$, with positive rank and $\Delta(v) \geq 0$, we
 define two characteristic curves on the $(\alpha, \beta)$-plane:
 
 \begin{align*}
-	V_v &\colon \chern_1^{\alpha, \beta}(v) = 0 \\
-	\Theta_v &\colon \chern_2^{\alpha, \beta}(v) = 0
+	V_v &\colon \:\: \chern_1^{\alpha, \beta}(v) = 0 \\
+	\Theta_v &\colon \:\: \chern_2^{\alpha, \beta}(v) = 0
 \end{align*}
 \end{definition}
 
@@ -366,26 +366,49 @@ The following facts can be deduced from the formulae for $\chern_i^{\alpha, \bet
 as well as the restrictions on $v$, when $\chern_0(v)=0$ and $\chern_1(v)>0$:
 
 
-\begin{minipage}{0.49\textwidth}
+\begin{minipage}{0.59\textwidth}
 \begin{itemize}
 	\item $V_v = \emptyset$
 	\item $\Theta_v$ is a vertical line at $\beta=\frac{D}{C}$
 		where $v=\left(0,C\ell,D\ell^2\right)$
 \end{itemize}
 \end{minipage}
-\begin{minipage}{0.49\textwidth}
+\hfill
+\begin{minipage}{0.39\textwidth}
 	\sageplot[width=\textwidth]{Theta_v_plot}
 	%\caption{$\Delta(v)>0$}
 	%\label{fig:charact_curves_rank0}
 \end{minipage}
 \end{fact}
 
+We can view the characteristic curves for $\left(0,C\ell, D\ell^2\right)$ with $C>0$ as
+the limiting behaviour of those of $\left(\varepsilon, C\ell, D\ell^2\right)$.
+Indeed:
+\begin{align*}
+	\mu\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C}{\varepsilon} &\longrightarrow +\infty
+	\\
+	\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 $\Theta_v^-$:
+\begin{align*}
+	\beta_{-}\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C - \sqrt{C^2-2D\varepsilon}}{\varepsilon}
+	&\longrightarrow \frac{D}{C}
+	\\
+	\text{as} \:\: 0<\varepsilon &\longrightarrow 0
+	&\text{(via L'H\^opital)}
+\end{align*}
+
+So we can view $\Theta_v^-$ as approaching the vertical line that $\Theta_v$  becomes.
+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.
 
 
 \subsection{Relevance of $V_v$}
 \label{subsect:relevance-of-V_v}
 
-By definition of the first tilt $\firsttilt\beta$, objects of Chern character
+For the positive rank case, by definition of the first tilt $\firsttilt\beta$, objects of Chern character
 $v$ can only be in $\firsttilt\beta$ on the left of $V_v$, where
 $\chern_1^{\alpha,\beta}(v)>0$, and objects of Chern character $-v$ can only be
 in $\firsttilt\beta$ on the right, where $\chern_1^{\alpha,\beta}(-v)>0$. In
@@ -396,6 +419,11 @@ Because of this, when using these characteristic curves, only positive ranks are
 considered, as negative rank objects are implicitly considered on the right hand
 side of $V_v$.
 
+In the rank zero case, this still applies if we consider $V_v$ to be
+`infinitely to the right' ($\mu(v) = +\infty$). Precisely, Gieseker semistable
+coherent sheaves $E$ of Chern character $v$ are contained in
+$\firsttilt{\beta}$ for all $\beta$
+
 
 
 \subsection{Relevance of $\Theta_v$}
@@ -406,7 +434,7 @@ $(\alpha$-$\beta)$-half-plane into regions where the signs of tilt slopes of
 objects of Chern character $v$ (or $-v$) are fixed. Secondly, it gives more of a
 fixed target for some $u=(r,c\ell,d\frac{1}{2}\ell^2)$ to be a
 pseudo-semistabilizer of $v$, in the following sense: If $(\alpha,\beta)$, is on
-$\Theta_v$, then for any $u$, $u$ is a pseudo-semistabilizer of $v$ iff
+$\Theta_v$, then for any $u$, $u$ can only be a pseudo-semistabilizer of $v$ if
 $\nu_{\alpha,\beta}(u)=0$, and hence $\chern_2^{\alpha, \beta}(u)=0$. In fact,
 this allows us to use the characteristic curves of some $v$ and $u$ (with
 $\Delta(v), \Delta(u)\geq 0$ and positive ranks) to determine the location of