diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index 06fd0f65a61cf0a5fccdaf316fbab77838b33a47..39571718277b7128ca951e83404f6d0f2617a00c 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -645,7 +645,7 @@ idea will be pursued in Subsection \ref{subsec:bounds-on-semistab-rank-prob-1}.
 \ref{problem:problem-statement-1}}
 \label{subsec:bounds-on-semistab-rank-prob-1}
 
-As discussed at the end of subsection \ref{subsubsect:all-bounds-on-d-prob1}
+As discussed at the end of Subsection \ref{subsubsect:all-bounds-on-d-prob1}
 (and illustrated in Figure \ref{fig:problem1:d_bounds_xmpl_gnrc_q}),
 there are no solutions $u$ to problem \ref{problem:problem-statement-1}
 with large $r=\chern_0(u)$, since the lower bound on $d=\chern_2(u)$ is larger
diff --git a/tex/characteristic-curves.tex b/tex/characteristic-curves.tex
index 2da28b8fb3b6533fbffa38b3a4b166f1a7c8eb64..e1511bce37871adcf2c46dd8b6117ed6da37dd79 100644
--- a/tex/characteristic-curves.tex
+++ b/tex/characteristic-curves.tex
@@ -23,7 +23,7 @@ define two characteristic curves on the $(\alpha, \beta)$-plane:
 These characteristic curves for a Chern character $v$ with $\Delta(v)\geq0$ are
 not affected by flipping the sign of $v$ so it's only necessary to consider
 non-negative rank.
-As discussed in subsection \ref{subsect:relevance-of-V_v}, making this choice
+As discussed in Subsection \ref{subsect:relevance-of-V_v}, making this choice
 has Gieseker stable coherent sheaves appearing in the heart of the stability
 condition $\firsttilt{\beta}$ as we move `left' (decreasing $\beta$).
 
@@ -91,7 +91,7 @@ degenerate_characteristic_curves
 	In particular, this means $\beta_\pm(v)$ are the two roots of the quadratic
 	equation $\chern_2^{\beta}(v)=0$.
 
-	This definition will be extended to the rank 0 case in definition \ref{dfn:beta_-_rank0}.
+	This definition will be extended to the rank 0 case in Definition \ref{dfn:beta_-_rank0}.
 \end{definition}
 
 
@@ -196,7 +196,7 @@ the pseudo-wall where $u$ pseudo-semistabilizes $v$. This is done by finding the
 intersection of $\Theta_v$ and $\Theta_u$, the point $(\beta,\alpha)$ where
 $\nu_{\alpha,\beta}(u)=\nu_{\alpha,\beta}(v)=0$, and a pseudo-wall point on
 $\Theta_v$, and hence the apex of the circular pseudo-wall with centre $(\beta,0)$
-(as per subsection \ref{subsect:bertrams-nested-walls}).
+(as per Subsection \ref{subsect:bertrams-nested-walls}).
 
 
 \subsection{Bertram's Nested Wall Theorem}
diff --git a/tex/computing-solutions.tex b/tex/computing-solutions.tex
index 0efaf3960c62f70fa1e36777a8e5acb615d071d4..f28d9a36f6fb855a76a1034c18dd4c8f360b48cb 100644
--- a/tex/computing-solutions.tex
+++ b/tex/computing-solutions.tex
@@ -21,7 +21,7 @@ Recalling Consequence 2 of Lemma \ref{lem:pseudo_wall_numerical_tests}, we can
 iterate through the possible values of $\mu(u)=\frac{c}{r}$ taking a decreasing
 sequence of all fractions between $\mu(v)$ and $\beta_{-}$, whos denominators
 are no large than $r_{\mathrm{max}}$ (giving a finite sequence). This can be done with
-Farey sequences \cite[chapter 6]{alma994504533502466}, for which there exist
+Farey sequences \cite[Chapter 6]{alma994504533502466}, for which there exist
 formulae to generate.
 
 These $\mu(u)$ values determine pairs $r,c$ up to multiples, we can then take
@@ -248,7 +248,7 @@ producing the results marginally faster.
 Note that this program patched with Theorem \ref{thm:loose-bound-on-r} will be
 using the same bound as was used in the previously existing program
 \cite{SchmidtGithub2020}. However the difference of performance can be of
-several orders of magnitude as illustrated in the table in section
+several orders of magnitude as illustrated in the table in Section
 \ref{table:bench-schmidt-vs-nay}.
 This will be attributed to the difference in programming language and algorithm,
 the latter having already been discussed in that same section.