diff --git a/tex/setting-and-problems.tex b/tex/setting-and-problems.tex
index da5240c3cb77751d4286d66809aa9382ac03beb1..03efa6117a6ccf35d83bf0e072532c52465bde37 100644
--- a/tex/setting-and-problems.tex
+++ b/tex/setting-and-problems.tex
@@ -1,28 +1,6 @@
-\section{Setting and Definitions: Clarifying `pseudo'}
-
-%\begin{definition}[Twisted Chern Character]
-%\label{sec:twisted-chern}
-%For a given $\beta$, define the twisted Chern character as follows.
-%\[\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)\]
-%\noindent
-%Component-wise, this is:
-%\begin{align*}
-% \chern^\beta_0(E) &= \chern_0(E)
-%\\
-% \chern^\beta_1(E) &= \chern_1(E) - \beta \chern_0(E)
-%\\
-% \chern^\beta_2(E) &= \chern_2(E) - \beta \chern_1(E) + \frac{\beta^2}{2} \chern_0(E)
-%\end{align*}
-%where $\chern_i$ is the coefficient of $\ell^i$ in $\chern$.
-%
-%% TODO I think this^ needs adjusting for general Surface with $\ell$
-%\end{definition}
-%
-%$\chern^\beta_1(E)$ is the imaginary component of the central charge
-%$\centralcharge_{\alpha,\beta}(E)$ and any element of $\firsttilt\beta$
-%satisfies $\chern^\beta_1 \geq 0$.
-
-Throughout this article, as noted in the introduction, we will be exclusively
+\section{Definitions: Clarifying `pseudo'}
+
+Throughout this Part, as noted in the introduction, we will be exclusively
working over surfaces $X$ with Picard rank 1, with a choice of ample line bundle
$L$ such that $\ell\coloneqq c_1(L)$ generates $NS(X)$.
We take $m\coloneqq \ell^2$ as this will be the main quantity which will
@@ -50,13 +28,21 @@ affect the results.
\item $0 \leq \chern_1^{\beta}(u) \leq \chern_1^{\beta}(v)$
\end{itemize}
+\end{definition}
+\begin{remark}
Note $u$ does not need to be a Chern character of an actual sub-object of some
object in the stability condition's heart with Chern character $v$.
-\end{definition}
-At this point, and in this document, we do not care about whether
-pseudo-semistabilisers are even Chern characters of actual elements of
-$\bddderived(X)$, some other sources may have this extra restriction too.
+ Other sources may also have extra conditions such as the integrality of Euler
+ characteristics or the non-triviality of certain extension groups.
+ These conditions depend on the Todd class of the variety in question and could
+ be considered but are left out for now as they do not have a great impact on
+ the finiteness of pseudo-walls.
+ In the case of a principally polarised abelian surface, the main example in
+ this Thesis, the Euler characteristic condition is vacuous and the extension
+ group condition eliminates possibities with lower rank, and often none at all
+ for small values of $\chern_0(v)$.
+\end{remark}
Later, Chern characters will be written $(r,c\ell,d\ell^2)$ because operations
(such as multiplication) are more easily defined in terms of the coefficients of
@@ -130,15 +116,17 @@ $d \in \frac{1}{\lcm(m,2)}\ZZ$.
\end{proof}
-\subsection{Characteristic Curves for Pseudo-semistabilisers}
+\section{Characteristic Curves for Pseudo-semistabilisers}
-These characteristic curves introduced are convenient tools to think about the
+These characteristic curves introduced in
+subsection \ref{subsec:charact-curves}
+are convenient tools to think about the
numerical conditions that can be used to test for pseudo-semistabilisers, and
for solutions to the problems
(\ref{problem:problem-statement-1},\ref{problem:problem-statement-2})
-tackled in this article (to be introduced later).
-In particular, problem (\ref{problem:problem-statement-1}) will be translated to
-a list of numerical inequalities on it's solutions $u$.
+tackled in this Part (to be introduced later).
+In particular, these problems will be translated to
+a list of numerical inequalities on its solutions $u$.
% ref to appropriate Lemma when it's written
The next Lemma is a key to making this translation and revolves around the
@@ -148,7 +136,8 @@ semistabilising sequence.
\begin{lemma}[Numerical tests for left-wall pseudo-semistabilisers]
\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 (and $\chern_1(v)>0$ if rank 0).
+\Delta(u)\geq 0$, and $v$ is positive, with one of $\chern_0(v)$ or
+$\chern_1(v)$ non-zero.
Let $P$ be a point on $\Theta_v^-$.
\noindent
@@ -257,6 +246,7 @@ Finally, consequence 4 along with $P$ being to the left of $V_u$ implies
$\nu_P(u) > 0$ giving supposition b.
The case with rank 0 can be handled the same way.
+% TODO expand this case too
\end{proof}
@@ -348,9 +338,11 @@ typically more familiar chamber
(the stable objects of Chern character $v$ in the outside chamber will only be
Gieseker stable sheaves).
-Also note that this last restriction does not remove any pseudo-walls found,
-and if we do want to recover `outwards' semistabilisers, we can simply take
-$v-u$ for each solution $u$ of the problem.
+\begin{remark}
+ Also note that this last restriction does not remove any pseudo-walls found,
+ and if we do want to recover `outwards' semistabilisers, we can simply take
+ $v-u$ for each solution $u$ of the problem.
+\end{remark}
\begin{problem}[all `left' pseudo-walls]
@@ -381,7 +373,7 @@ Note also that the condition on $\beta_-(v)$ always holds for $v$ rank 0.
The problems introduced in this section are phrased in the context of stability
conditions. However, these can be reduced down completely to purely numerical
-problem with the help of Lemma \ref{lem:pseudo_wall_numerical_tests}.
+problem using Lemma \ref{lem:pseudo_wall_numerical_tests}.
\begin{lemma}[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
\label{lem:num_test_prob1}