diff --git a/main.tex b/main.tex
index e64982bd5be5277d9ba16d2cd5b3888fb686b36d..5e0cd804394cfe3287a93289a47f6c392c78c011 100644
--- a/main.tex
+++ b/main.tex
@@ -93,7 +93,7 @@ stability space. These are where there is some stable object $F$ of $v$ which
 has a subobject who's slope overtakes the slope of $v$, making $F$ unstable
 after crossing the wall.
 
-% NOTE: SURFACE SPACIALIZATION
+% NOTE: SURFACE SPECIALIZATION
 % (come back to these when adjusting to general Picard rank 1)
 In this document we concentrate on two surfaces: Principally polarized abelian
 surfaces and the projective surface $\PP^2$. Although this can be generalised
@@ -154,8 +154,13 @@ on $\chern_0(E)$ of potential destabilizers $E$ of $F$.
 \section{Characteristic Curves of Stability Conditions Associated to Chern
 Characters}
 
+Throughout this article, as noted in the introduction, we will be exclusively
+working over one of the following two surfaces: principally polarized abelian
+surfaces and $\PP^2$.
+
 \begin{dfn}[Pseudo-semistabilizers]
-	Given a Chern Character $v$ on a Picard rank 1 surface, and a given stability
+% NOTE: SURFACE SPECIALIZATION
+	Given a Chern Character $v$, and a given stability
 	condition $\sigma_{\alpha,\beta}$,
 	a pseudo-semistabilizing $u$ is a `potential' Chern character:
 	\[
@@ -186,13 +191,15 @@ pseudo-semistabilizers are even Chern characters of actual elements of
 $\bddderived(X)$, some other sources may have this extra restriction too.
 
 \begin{lemma}[ Sanity check for Pseudo-semistabilizers ]
-	Given a Picard rank 1 surface, and a given stability
+% NOTE: SURFACE SPECIALIZATION
+	Given a stability
 	condition $\sigma_{\alpha,\beta}$,
 	if $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing sequence in
 	$\firsttilt\beta$ for $F$.
 	Then $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$
 \end{lemma}
 
+% NOTE: SURFACE SPECIALIZATION
 Considering the stability conditions with two parameters $\alpha, \beta$ on
 Picard rank 1 surfaces.
 We can draw 2 characteristic curves for any given Chern character $v$ with
@@ -658,6 +665,7 @@ rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
 
 \section{B.Schmidt's Wall Finding Method}
 
+% NOTE: SURFACE SPECIALIZATION
 The proof for the previous theorem was hinted at in
 \cite{SchmidtBenjamin2020Bsot}, but the value appears explicitly in
 \cite{SchmidtGithub2020}. The latter reference is a SageMath \cite{sagemath}