diff --git a/tex/content.tex b/tex/content.tex
index eed348e9639809eac921e735ab3ef1b85d16adc4..5d888bd245b283bbbaa66c59df4de42f62e4c2ef 100644
--- a/tex/content.tex
+++ b/tex/content.tex
@@ -217,7 +217,7 @@ 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
 $\Delta(v) \geq 0$ and positive rank.
-These are given by the equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$.
+These are given by the Equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$.
 
 \begin{definition}[Characteristic Curves $V_v$ and $\Theta_v$]
 Given a Chern character $v$, with positive rank and $\Delta(v) \geq 0$, we
@@ -300,7 +300,7 @@ degenerate_characteristic_curves
 	\]
 	\noindent
 	In particular, this means $\beta_\pm(v)$ are the two roots of the quadratic
-	equation $\chern_2^{\beta}(v)=0$.
+	Equation $\chern_2^{\beta}(v)=0$.
 
 	This definition will be extended to the rank 0 case in definition \ref{dfn:beta_-_rank0}.
 \end{definition}
@@ -1028,7 +1028,7 @@ from plots_and_expressions import bgmlv2_d_ineq
 \begin{sagesilent}
 from plots_and_expressions import bgmlv2_d_upperbound_terms
 \end{sagesilent}
-Viewing equation \ref{eqn-bgmlv2_d_upperbound} as a lower bound for $d$ in term
+Viewing Equation \ref{eqn-bgmlv2_d_upperbound} as a lower bound for $d$ in term
 of $r$ again, there is a constant term
 $\sage{bgmlv2_d_upperbound_terms.const}$,
 a linear term
@@ -1077,7 +1077,7 @@ from plots_and_expressions import bgmlv3_d_upperbound_terms
 For $r=R$, $\Delta(v-u)\geq 0$ is always true, and for $r<R$ it gives a lower
 bound on $d$, but it is weaker than the one given by the lower bound in
 subsubsection \ref{subsect-d-bound-radiuscond}.
-Viewing the right hand side of equation \ref{eqn-bgmlv3_d_upperbound}
+Viewing the right hand side of Equation \ref{eqn-bgmlv3_d_upperbound}
 as a function of $r$, the linear and constant terms almost match up with the
 ones in the previous section, up to the 
 $\chern_2^{\beta}(v)$ term.
@@ -1251,7 +1251,7 @@ bounds do not share the same assymptote as the lower bound
 
 Notice that as a function in $r$, the linear term in 
 equation \ref{eqn:prob1:radiuscond} is strictly greater than
-those in equations \ref{eqn:prob1:bgmlv2}
+those in Equations \ref{eqn:prob1:bgmlv2}
 and \ref{eqn:prob1:bgmlv3}. This is because $r$, $R$
 and $\chern_2^B(v)$ are all strictly positive:
 \begin{itemize}
@@ -1307,16 +1307,16 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
 
 \begin{proof}
 	Recall that $d\coloneqq\chern_2(u)$ has two upper bounds in terms of $r$: in
-	equations \ref{eqn:prob1:bgmlv2} and \ref{eqn:prob1:bgmlv3};
-	and one lower bound: in equation \ref{eqn:prob1:radiuscond}.
+	Equations \ref{eqn:prob1:bgmlv2} and \ref{eqn:prob1:bgmlv3};
+	and one lower bound: in Equation \ref{eqn:prob1:radiuscond}.
 
-	Solving for the lower bound in equation \ref{eqn:prob1:radiuscond} being
-	less than the upper bound in equation \ref{eqn:prob1:bgmlv2} yields:
+	Solving for the lower bound in Equation \ref{eqn:prob1:radiuscond} being
+	less than the upper bound in Equation \ref{eqn:prob1:bgmlv2} yields:
 	\begin{equation}
 	r<\sage{problem1.positive_intersection_bgmlv2}
 	\end{equation}
 
-	Similarly, but with the upper bound in equation \ref{eqn:prob1:bgmlv3}, gives:
+	Similarly, but with the upper bound in Equation \ref{eqn:prob1:bgmlv3}, gives:
 	\begin{equation}
 	r<\sage{problem1.positive_intersection_bgmlv3}
 	\end{equation}
@@ -1446,7 +1446,7 @@ Hence any corresponding $r$ cannot be a rank of a
 pseudo-semistabilizer for $v$.
 
 To avoid this, we must have,
-considering equations
+considering Equations
 \ref{eqn:bgmlv2_d_bound_betamin},
 \ref{eqn:bgmlv3_d_bound_betamin},
 \ref{eqn:radiuscond_d_bound_betamin}.