Add discussion around Bertram's nested walls and statement of numerical conditions for walls

main.tex

@@ -27,6 +27,7 @@
 \newtheorem{corrolary}{Corrolary}[section]
 \newtheorem{lemmadfn}{Lemma/Definition}[section]
 \newtheorem{dfn}{Definition}[section]
+\newtheorem{lemma}{Lemma}[section]
 
 \begin{document}
 
@@ -199,7 +200,7 @@ If $(\alpha,\beta)$, is on the hyperbola $\chern_2^{\alpha, \beta}(v)=0$, then
 for any $u$, $u$ is a pseudo-semistabilizer of $v$
 iff $\mu_{\alpha,\beta}(u)=0$, and hence $\chern_2^{\alpha, \beta}(u)=0$.
 In fact, this allows us to use the characteristic curves of some $v$ and $u$
-(with $\Delta(v)\geq 0$, $\Delta(u)\geq 0$ and positive ranks) to determine the
+(with $\Delta(v), \Delta(u)\geq 0$ and positive ranks) to determine the
 location of the pseudo-wall where $u$ pseudo-semistabilizes $v$.
 %TODO ref forwards
 
@@ -209,8 +210,48 @@ consequence of $\Delta(v)\geq 0$. Furthermore the assymptotes are angled at $\pm
 45^\circ$, crossing through the base of the first characteristic curve
 $\chern_1^{\alpha,\beta}=0$ (vertical line).
 
+\subsection{Bertram's nested wall theorem}
 
+Although Bertram's nested wall theorem can be proved more directly, it's also
+important for the content of this document to understand the connection with
+these characteristic curves.
+Emanuele Macri noticed in (TODO ref) that any circular wall of $v$ reaches a critical
+point on the second critical curve for $v$ ($\chern_2^{\alpha, \beta}(v)=0$),
+this is a consequence of
+$\frac{\delta}{\delta\beta} \chern_2^{\alpha,\beta} = -\chern_1^{\alpha,\beta}$.
+This fact, along with the hindsight knowledge that non-vertical walls are
+circles with centers on the $\beta$-axis, gives an alternative view to see that
+the circular walls must be nested and non-intersecting.
 
+\subsection{Characteristic curves for pseudo-semistabilizers}
+
+\begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
+Let $v$ and $u$ be Chern characters with $\Delta(v), \Delta(u)\geq 0$ and
+positive ranks.
+
+Suppose that $u$ gives rise to a pseudo-wall for $v$, left of the characteristic
+vertical line $\chern_1^{\alpha,\beta}(v)=0$ and containing a fixed point $p$ in
+it's interior.
+To target all left-walls, $p$ can be chosen as the base of the left branch of
+the hyperbola $\chern_2^{\alpha,\beta}(v)=0$.
+Suppose further that this happens in a way such that $u$ destabilizes $v$ going
+`inwards', that is,
+$\nu_{\alpha,\beta}(u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
+$\nu_{\alpha,\beta}(u)>\nu_{\alpha,\beta}(v)$ inside.
+
+
+\noindent
+Then we have the following:
+\begin{itemize}
+	\item The pseudo-wall is left of $u$'s vertical characteristic line
+		(if this is a real wall then $v$ is being semistabilized by a positive rank object)
+	\item $\mu(u)<\mu(v)$, i.e., $u$'s vertical characteristic line is left of $v$'s vertical
+		characteristic line
+	\item $\chern_2^{p}(u)>0$
+\end{itemize}
+Furthermore, only the last two of these consequences are sufficient to recover
+all of the suppositions above.
+\end{lemma}