Extend numerical formulations to rank 0

main.tex

@@ -506,7 +506,8 @@ semistabilizing sequence.
 \begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
 \label{lem:pseudo_wall_numerical_tests}
 Let $v$ and $u$ be Chern characters with $\Delta(v),
-\Delta(u)\geq 0$, and $v$ has non-negative rank. Let $P$ be a point on $\Theta_v^-$.
+\Delta(u)\geq 0$, and $v$ has non-negative rank (and $\chern_1(v)>0$ if rank 0).
+Let $P$ be a point on $\Theta_v^-$.
 
 \noindent
 The following conditions:
@@ -635,7 +636,8 @@ are trying to solve for.
 \begin{problem}[sufficiently large `left' pseudo-walls]
 \label{problem:problem-statement-1}
 
-Fix a Chern character $v$ with non-negative rank, and $\Delta(v) \geq 0$.
+Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
+and $\Delta(v) \geq 0$.
 The goal is to find all pseudo-semistabilizers $u$
 which give circular pseudo-walls containing some fixed point
 $P\in\Theta_v^-$.
@@ -670,8 +672,8 @@ $v-u$ for each solution $u$ of the problem.
 \begin{problem}[all `left' pseudo-walls]
 \label{problem:problem-statement-2}
 
-Fix a Chern character $v$ with non-negative rank, $\Delta(v) \geq 0$,
-and $\beta_{-}(v) \in \QQ$.
+Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
+$\Delta(v) \geq 0$, 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}
@@ -697,7 +699,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 
 \begin{lemma}[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
 	\label{lem:num_test_prob1}
-	Given a Chern character $v$ with positive rank and $\Delta(v) \geq 0$,
+	Given a Chern character $v$ with non-negative rank
+	(with $\chern_1(v)>0$ if rank 0)
+	and $\Delta(v) \geq 0$,
 	and a choice of point $P$ on $\Theta_v^-$.
 	Solutions $u=(r,c\ell,\frac{e}{\lcm(m,2)}\ell^2)$
 	to problem \ref{problem:problem-statement-1}.
@@ -718,7 +722,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 \end{lemma}
 
 \begin{proof}
-	Consider the context of $v$ being a Chern character with positive rank and
+	Consider the context of $v$ being a Chern character with non-negative rank
+	(and $\chern_1(v)>0$ if rank 0)
+	and
 	$\Delta \geq 0$, and $u$ being a Chern character with $\Delta(u) \geq 0$.
 	Lemma \ref{lem:pseudo_wall_numerical_tests} gives that the remaining
 	conditions for $u$ being a solution to problem