Extend problem statement to rank 0

main.tex

@@ -372,6 +372,7 @@ degenerate_characteristic_curves
 
 
 \subsubsection{Rank Zero Case}
+\label{subsubsect:rank-zero-case-charact-curves}
 
 \begin{sagesilent}
 from rank_zero_case import Theta_v_plot
@@ -634,7 +635,7 @@ are trying to solve for.
 \begin{problem}[sufficiently large `left' pseudo-walls]
 \label{problem:problem-statement-1}
 
-Fix a Chern character $v$ with positive rank, and $\Delta(v) \geq 0$.
+Fix a Chern character $v$ with non-negative rank, and $\Delta(v) \geq 0$.
 The goal is to find all pseudo-semistabilizers $u$
 which give circular pseudo-walls containing some fixed point
 $P\in\Theta_v^-$.
@@ -669,19 +670,25 @@ $v-u$ for each solution $u$ of the problem.
 \begin{problem}[all `left' pseudo-walls]
 \label{problem:problem-statement-2}
 
-Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
+Fix a Chern character $v$ with non-negative rank, $\Delta(v) \geq 0$,
 and $\beta_{-}(v) \in \QQ$.
 The goal is to find all pseudo-semistabilizers $u$ which give circular
 pseudo-walls on the left side of $V_v$.
 \end{problem}
 
 This is a specialization of problem (\ref{problem:problem-statement-1})
-with the choice $P=(\beta_{-},0)$.
+with the choice $P=(\beta_{-},0)$, the point where $\Theta_v^-$ meets the
+$\beta$-axis.
 This is because all circular walls left of $V_v$ intersect $\Theta_v^-$ (once).
 The $\beta_{-}(v) \in \QQ$ condition is to ensure that there are finitely many
 solutions. As mentioned in the introduction (\ref{sec:intro}), this is known,
 however this will also be proved again in passing in this article.
 
+This description still holds for the case of rank 0 case if we consider $V_v$ to
+be infinitely far to the right
+(see section \ref{subsubsect:rank-zero-case-charact-curves}).
+Note also that the $\beta_-(v)$ condition always holds for $v$ rank 0.
+
 \subsection{Numerical Formulations of the Problems}
 
 The problems introduced in this section are phrased in the context of stability