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Commit 1e395091 authored by Luke Naylor's avatar Luke Naylor
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Generalise main theorems to other picard rank 1 surfaces

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......@@ -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}
......
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