Complete corrections of part II

tex/bounds-on-semistabilisers.tex

@@ -81,8 +81,8 @@ pseudo-semistabilisers for tilt stability.
 	currently considered restrictions, is unbounded.
 	This is where the rationality of $\beta_{-}$ comes in.
 	If $\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$,
-	then $\chern^{\beta_-}_2(u) \in \frac{1}{\lcm(m,2n^2)}\ZZ$.
-	In particular, since $\chern_2^{\beta_-}(u) > 0$ we must also have
+	then we must have $\chern^{\beta_-}_2(u) \in \frac{1}{\lcm(m,2n^2)}\ZZ$.
+	In particular, since $\chern_2^{\beta_-}(u) > 0$ we have
 	$\chern^{\beta_-}_2(u) \geq \frac{1}{\lcm(m,2n^2)}$, which then in turn gives a
 	bound for the rank of $u$:
 
@@ -90,11 +90,11 @@ pseudo-semistabilisers for tilt stability.
 		\chern_0(u)
 		 & \leq \frac{\chern^{\beta_-}_1(u)^2}{2\chern^{\beta_{-}}_2(u)} \nonumber \\
 		 & \leq \frac{\lcm(m,2n^2) \chern^{\beta_-}_1(u)^2}{2} \nonumber           \\
-		 & = \frac{mn^2 \chern^{\beta_-}_1(u)^2}{\gcd(m,2n^2)}
+		 & = \frac{mn^2 \chern^{\beta_-}_1(u)^2}{\gcd(m,2n^2)}.
 		\label{proof:first-bound-on-r}
 	\end{align}
 	\noindent
-	Which we can then immediately bound using Equation \eqref{eqn-tilt-cat-cond}.
+	This can then immediately be bound using Equation \eqref{eqn-tilt-cat-cond}.
 	Alternatively, given that
 	$\chern_1^{\beta_{-}}(u)$, $\chern_1^{\beta_{-}}(v)\in\frac{1}{n}\ZZ$,
 	we can tighten this bound on $\chern_1^{\beta_{-}}(u)$ given by that equation to:
@@ -142,7 +142,7 @@ pseudo-semistabilisers for tilt stability.
 	Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
 	tilt semistabilisers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
 	However, when computing all tilt semistabilisers for $v$ on $\PP^2$, the maximum
-	rank that appears turns out to be $\sage{round(extravagant.actual_rmax, 1)}$.
+	rank that appears turns out to be $\sage{round(extravagant.actual_rmax)}$.
 \end{example}
 
 
@@ -180,8 +180,7 @@ of travel.
 		\qquad
 		\text{and}
 		\qquad
-		\chern_0(u) > \frac{q}{\mu(v) - \beta_0}.
-		\nonumber
+		\chern_0(u) > \frac{q}{\mu(v) - \beta_0}\cdot
 	\end{equation}
 
 	\noindent
@@ -212,7 +211,7 @@ of travel.
 	$u=(r,c\ell,d\ell^2)$ with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$ satisfying Equation
 	\eqref{lem:eqn:cond-for-fixed-q} is in fact a solution provided the remaining numerical conditions
 	1, 2, 3 and 6 are satisfied.
-	This is in essence the second part of the Lemma.
+	This is in essence the second part of the lemma.
 \end{proof}
 
 \begin{corollary}

tex/computing-solutions.tex

@@ -247,7 +247,7 @@ This could be due to a range of potential reasons:
 For relatively small Chern characters (as those appearing in examples so far),
 the difference in performance between the program \cite{NaylorRust2023} when
 patched with the results of the different theorems above, do not show any
-significant difference in performance. The earlier, weaker theorems occasionally
+significant difference in performance, with the earlier, weaker theorems occasionally
 producing the results marginally faster.
 
 Note that this program patched with Theorem \ref{thm:loose-bound-on-r} will be
@@ -271,8 +271,8 @@ indicators of the size of the bounds on the pseudo-semistabiliser ranks.
 	\caption{
 		Comparing the performance of program \cite{NaylorRust2023}
 		with different patches corresponding to the results of Theorems
-		\ref{thm:loose-bound-on-r}
-		\ref{thm:rmax_with_uniform_eps}
+		\ref{thm:loose-bound-on-r},
+		\ref{thm:rmax_with_uniform_eps},
 		\ref{thm:rmax_with_eps1}
 		when computing solutions to Problem \ref{problem:problem-statement-2}
 		for $(45,54\ell,-41\frac{\ell^2}{2})$
@@ -304,6 +304,6 @@ versus the average looser.
 However, the actual ratio in the benchmark shown in Figure \ref{fig:benchmark}
 between the two instances of the program patched with the two corresponding
 bounds is around 0.6 instead.
-Not as good as the improvement on the bound, however still not insignificant
+This is not as good as the improvement on the bound, however it is still not insignificant
 for examples with larger `$n$'-value, where the execution time will
 potentially be in the order of minutes, or even hours.