Generalise main theorems to other picard rank 1 surfaces

main.tex

@@ -1514,8 +1514,7 @@ $\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.
 \end{proof}
 
 \begin{sagesilent}
-from plots_and_expressions import r_upper_bound1, r_upper_bound2
-	
+from plots_and_expressions import main_theorem2
 \end{sagesilent}
 \begin{theorem}[Bound on $r$ \#3]
 \label{thm:rmax_with_eps1}
@@ -1533,8 +1532,8 @@ from plots_and_expressions import r_upper_bound1, r_upper_bound2
 	\begin{align*}
 		\min
 		\left(
-			\sage{r_upper_bound1.rhs()}, \:\:
-			\sage{r_upper_bound2.rhs()}
+			\sage{main_theorem2.r_upper_bound1}, \:\:
+			\sage{main_theorem2.r_upper_bound2}
 		\right)
 	\end{align*}
 	\egroup
@@ -1545,6 +1544,51 @@ from plots_and_expressions import r_upper_bound1, r_upper_bound2
 	$r \equiv \aa^{-1}b_q \pmod{n}$.
 \end{theorem}
 
+Although the general form of this bound is quite complicated, it does simplify a
+lot when $m$ is small.
+
+\begin{sagesilent}
+from plots_and_expressions import main_theorem2_corollary
+\end{sagesilent}
+\begin{corollary}[Bound on $r$ \#3 on $\PP^2$ and Principally polarized abelian surfaces]
+\label{thm:rmax_with_eps1}
+	Suppose we are working over $\PP^2$ or a principally polarized abelian surface
+	(or any other surfaces with $m=1$ or $2$).
+	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-semistabilizers $u$ 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:
+
+	\bgroup
+	\def\kappa{k_{v,q}}
+	\def\psi{\chern_1^{\beta}(F)}
+	\begin{align*}
+		\min
+		\left(
+			\sage{main_theorem2_corollary.r_upper_bound1}, \:\:
+			\sage{main_theorem2_corollary.r_upper_bound2}
+		\right)
+	\end{align*}
+	\egroup
+	Where $R = \chern_0(v)$ and $k_{v,q}$ is the least
+	$k\in\ZZ_{>0}$ satisfying
+	\begin{equation*}
+		k \equiv -\aa\bb
+		\pmod{n}
+	\end{equation*}
+
+	\noindent
+	Furthermore, if $\aa \not= 0$ then
+	$r \equiv \aa^{-1}b_q \pmod{n}$.
+\end{corollary}
+
+\begin{proof}
+This is a specialisation of theorem \ref{thm:rmax_with_eps1}, where we can
+drastically simplify the $\lcm$ and $\gcd$ terms by noting that $m$ divides both
+$2$ and $2n^2$, and that $a_v$ is coprime to $n$.
+\end{proof}
 
 \begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
 \label{exmpl:recurring-third}