Complete first pass of corrections on chapter 3

tex/setting-and-problems.tex

@@ -29,6 +29,12 @@ affect the results.
 	\end{itemize}
 
 \end{definition}
+\begin{remark}
+	We could introduce a slightly stronger definition including an extra condition on $e$
+	in terms of $r$ and $c$ to ensure that $u$ could arise from integral Chern classes.
+	However, this will not affect finiteness questions considered later and this also
+	condition turns out to be vacuous for principally polarised abelian surfaces.
+\end{remark}
 \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 $v$.
@@ -65,9 +71,10 @@ $d \in \frac{1}{\lcm(m,2)}\ZZ$.
 \label{lem:sanity-check-for-pseudo-semistabilizers}
 	Given a stability
 	condition $\sigma_{\alpha,\beta}$,
-	if $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilising sequence in
-	$\firsttilt\beta$ for $F$.
-	Then $\chern(E)$ is a pseudo-semistabiliser of $\chern(F)$
+	and a semistabilising sequence
+  $E\hookrightarrow F\twoheadrightarrow G$
+	in $\firsttilt\beta$ for $F$,
+	then $\chern(E)$ is a pseudo-semistabiliser of $\chern(F)$
 \end{lemma}
 
 \begin{proof}
@@ -368,8 +375,8 @@ 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-semistabilisers $u$
 which give circular pseudo-walls containing some fixed point
-$P\in\Theta_v^-$.
-With the added restriction that $u$ `destabilises' $v$ moving `inwards', that is,
+$P\in\Theta_v^-$
+with the added restriction that $u$ `destabilises' $v$ moving `inwards', that is,
 $\nu(u)>\nu(v)$ inside the circular pseudo-wall.
 \end{problem}
 This will give all pseudo-walls between the chamber corresponding to Gieseker
@@ -438,8 +445,8 @@ problem using Lemma \ref{lem:pseudo_wall_numerical_tests}.
 	and $\Delta(v) \geq 0$,
 	and a choice of point $P=(\alpha_0, \beta_0)$ on $\Theta_v^-$.
 	Solutions $u=(r,c\ell,d\ell^2)$
-	to Problem \ref{problem:problem-statement-1}.
-	Are precisely given by $r,c \in \ZZ$, $d \in \frac{1}{\lcm(m,2)}$
+	to Problem \ref{problem:problem-statement-1}
+	are precisely given by $r,c \in \ZZ$, $d \in \frac{1}{\lcm(m,2)}$
 	satisfying the following conditions:
 	\begin{enumerate}
 		\begin{multicols}{2}
@@ -479,8 +486,8 @@ problem using Lemma \ref{lem:pseudo_wall_numerical_tests}.
 	and $\Delta(v) \geq 0$,
 	such that $\beta_{-}\coloneqq\beta_{-}(v) \in \QQ$.
 	Solutions $u=(r,c\ell,d\ell^2)$
-	to Problem \ref{problem:problem-statement-2}.
-	Are precisely given by $r,c \in \ZZ$, $d\in\frac{1}{\lcm(m,2)}\ZZ$ satisfying
+	to Problem \ref{problem:problem-statement-2}
+	are precisely given by $r,c \in \ZZ$, $d\in\frac{1}{\lcm(m,2)}\ZZ$ satisfying
 	the following conditions:
 	\begin{enumerate}
 		\begin{multicols}{2}