Complete proof and misc in problems sect

main.tex

@@ -654,14 +654,13 @@ $v-u$ for each solution $u$ of the problem.
 
 Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
 and $\beta_{-}(v) \in \QQ$.
-The goal is to find all solutions $u$
-to problem \ref{problem:problem-statement-1} with the choice
-$P=(\beta_{-},0)$.
+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})
-which will give all circular pseudo-walls left of $V_v$.
-This is because all circular walls left of $V_v$ intersect $\Theta_v^-$.
+with the choice $P=(\beta_{-},0)$.
+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.
@@ -680,17 +679,22 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 	Are precisely given by integers $r,c,d$ satisfying the following conditions:
 	\begin{enumerate}
 		\item $r > 0$
-		\item	$\Delta(u) \geq 0$
-		\item	$\Delta(v-u) \geq 0$
+		\item $\Delta(u) \geq 0$
+		\item $\Delta(v-u) \geq 0$
 		\item $\beta(P)<\mu(u)=\frac{c}{r}<\mu(v)$
-		\item $\chern_1^{\beta(P)}(v-u)\geq0$
+		\item $\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$
 		\item $\chern_2^{P}(u)>0$
 	\end{enumerate}
 \end{lemma}
 
 \begin{proof}
-	% TODO complete
-	Use main lemma \ref{lem:pseudo_wall_numerical_tests} TODO
+	Consider the context of $v$ being a Chern character with positive rank 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
+	\ref{problem:problem-statement-1} are precisely equivalent to the
+	remaining conditions in this lemma.
+
 \end{proof}
 
 \begin{corrolary}[Numerical Tests for All `left' Pseudo-walls]
@@ -701,16 +705,17 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 	Are precisely given by integers $r,c,d$ satisfying the following conditions:
 	\begin{enumerate}
 		\item $r > 0$
-		\item	$\Delta(u) \geq 0$
-		\item	$\Delta(v-u) \geq 0$
+		\item $\Delta(u) \geq 0$
+		\item $\Delta(v-u) \geq 0$
 		\item $\beta(P)<\mu(u)=\frac{c}{r}<\mu(v)$
-		\item $\chern_1^{\beta_{-}}(v-u)\geq0$
+		\item $\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$
 		\item $\chern_2^{\beta_{-}}(u)>0$
 	\end{enumerate}
 \end{corrolary}
 
 \begin{proof}
 	This is a specialization of the previous lemma, using $P=(\beta_{-},0)$.
+
 \end{proof}