Iron out final theorem bound of chapter

tex/bounds-on-semistabilisers.tex

@@ -730,9 +730,9 @@ proof of Theorem
 
 
 \begin{sagesilent}
-from plots_and_expressions import main_theorem1
+from plots_and_expressions import main_theorem1, betamin_subs
 \end{sagesilent}
-\begin{theorem}[Bound on $r$ \#1]
+\begin{theorem}[Bound on $r$ \#1 for Problem \ref{problem:problem-statement-2}]
 \label{thm:rmax_with_uniform_eps}
 	Let $X$ be a smooth projective surface with Picard rank 1 and choice of ample
 	line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
@@ -747,8 +747,8 @@ from plots_and_expressions import main_theorem1
 	\begin{align*}
 		\min
 		\left(
-			\sage{main_theorem1.r_upper_bound1}, \:\:
-			\sage{main_theorem1.r_upper_bound2}
+			\sage{main_theorem1.r_upper_bound1.subs(betamin_subs)}, \:\:
+			\sage{main_theorem1.r_upper_bound2.subs(betamin_subs)}
 		\right)
 	\end{align*}
 	\noindent
@@ -760,8 +760,8 @@ Both $d$ and the lower bound in
 (Equation \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
 are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
 So, if any of the two upper bounds on $d$ come to within
-$\frac{1}{\lcm(m,2n^2)}$ of this lower bound, then there are no solutions for
-$d$.
+$\epsilon_v \coloneqq \frac{1}{\lcm(m,2n^2)}$ of this lower bound,
+then there are no solutions for $d$.
 Hence any corresponding $r$ cannot be a rank of a
 pseudo-semistabiliser for $v$.
 
@@ -778,8 +778,8 @@ assymptote_gap_condition1, assymptote_gap_condition2, k
 
 
 \begin{align}
-	&\sage{assymptote_gap_condition1.subs(k==1)} \\
-	&\sage{assymptote_gap_condition2.subs(k==1)}
+	\epsilon_v =&\sage{assymptote_gap_condition1.subs(k==1)} \\
+	\epsilon_v =&\sage{assymptote_gap_condition2.subs(k==1)}
 \end{align}
 
 \noindent
@@ -805,7 +805,7 @@ This is equivalent to:
 from plots_and_expressions import q_sol, bgmlv_v, psi
 \end{sagesilent}
 
-\begin{corollary}[Bound on $r$ \#2]
+\begin{corollary}[Global bound on $r$ \#2 for Problem \ref{problem:problem-statement-2}]
 \label{cor:direct_rmax_with_uniform_eps}
 	Let $X$ be a smooth projective surface with Picard rank 1 and choice of ample
 	line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
@@ -1042,29 +1042,37 @@ $\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.
 \begin{sagesilent}
 from plots_and_expressions import main_theorem2
 \end{sagesilent}
-\begin{theorem}[Bound on $r$ \#3]
+\begin{theorem}[Bound on $r$ \#3 for Problem \ref{problem:problem-statement-2}]
 \label{thm:rmax_with_eps1}
-	Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
-	rational and expressed in lowest terms.
-	Then the ranks $r$ of the pseudo-semistabilisers $u$ for $v$ with,
-	which are solutions to problem \ref{problem:problem-statement-2},
-	$\chern_1^\beta(u) = q = \frac{b_q}{n}$
-	are bounded above by the following expression:
+	Let $X$ be a smooth projective surface with Picard rank 1 and choice of ample
+	line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
+	$m\coloneqq\ell^2$.
+	Let $v$ be a fixed Chern character on this surface with positive rank
+	(or rank 0 and $c_1(v)>0$), and $\Delta(v)\geq 0$.
+	Then the ranks of the pseudo-semistabilisers $u$ for $v$,
+	which are solutions to Problem \ref{problem:problem-statement-2},
+	with $\chern_1^{\beta_{-}(v)}(u) = q$
+	are bounded above by the following expression.
 
 	\begin{align*}
 		\min
 		\left(
-			\sage{main_theorem2.r_upper_bound1}, \:\:
-			\sage{main_theorem2.r_upper_bound2}
+			\sage{main_theorem2.r_upper_bound1.subs(betamin_subs)}, \:\:
+			\sage{main_theorem2.r_upper_bound2.subs(betamin_subs)}
 		\right)
 	\end{align*}
-	Where $k_{v,q}$ is defined as in definition/Lemma \ref{lemdfn:epsilon_q},
+	Where $k_{v,q}$ is defined as in Definition/Lemma \ref{lemdfn:epsilon_q},
 	and $R = \chern_0(v)$
-
-	Furthermore, if $\aa \not= 0$ then
-	$r \equiv \aa^{-1}b_q \pmod{n}$.
 \end{theorem}
 
+\begin{proof}
+	Following the same proof as Theorem \ref{thm:rmax_with_uniform_eps},
+	$\epsilon_{v,q} = \frac{k_{v,q}}{\lcm(m, 2n^2)}$ can be used instead of
+	$\epsilon_{v} = \frac{1}{\lcm(m, 2n^2)}$ as it satisfies the same required
+	property, as per Definition/Lemma \ref{lemdfn:epsilon_q}.
+	
+\end{proof}
+
 Although the general form of this bound is quite complicated, it does simplify a
 lot when $m$ is small.