bounds-on-semistabilisers.tex

	with $k_{v,q}$ being the least $k\in\ZZ_{>0}$ satisfying
	\begin{equation*}
		k \equiv -\aa\bb \frac{m}{\gcd(m,2n^2)}
		\mod{\gcd\left(
			\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
			\frac{mn\aa}{\gcd(m,2n^2)}
		\right)}
	\end{equation*}

\end{lemmadfn}

\begin{remark}
    The quantity $m$ is determined by the variety, whereas $a_v$ and $n$ are determined by the Chern
    character $v$ for which we are trying to find pseudo-semistabilisers.
    So the $\gcd$ expression we are taking the modulus with respect to is considered
    constant in the context of the problem we are solving for
    (i.e. Problem \ref{problem:problem-statement-2}).
    However $b_q$ depends on the choice of $q$, that is the value of
    $\chern_1^{\beta_{-}(v)}(u)$ for which we are searching for solutions $u$, hence
    why $k_{v,q}$ is denoted to depend on $q$ on top of $v$ and the context of the problem.
\end{remark}

\begin{proof}

Consider the following sequence of logical implications.
The one-way implication follows from
$\aa r + \bb \equiv 0 \pmod{n}$,
and the final logical equivalence is just a simplification of the expressions.

\begin{align}
	\frac{ x }{ \lcm(m,2) }
	- \frac{
		(\aa r+2\bb)\aa
	}{
		2n^2
	}
	= \frac{ k }{ \lcm(m,2n^2) }
	\quad \text{for some } x \in \ZZ
	\span \span \span \span \span
	\label{eqn:finding_better_eps_problem}
\\ \nonumber
\\ \Leftrightarrow& &
	- (\aa r+2\bb)\aa
	\frac{\lcm(m,2n^2)}{2n^2}
	&\equiv k &&
	\nonumber
\\ &&&
	\mod \frac{\lcm(m,2n^2)}{\lcm(m,2)}
	\span \span \span
	\nonumber
\\ \Rightarrow& &
	- \bb\aa
	\frac{\lcm(m,2n^2)}{2n^2}
	&\equiv k &&
	\nonumber
\\ &&&
	\mod \gcd\left(
		\frac{\lcm(m,2n^2)}{\lcm(m,2)},
		\frac{n \aa \lcm(m,2n^2)}{2n^2}
	\right)
	\span \span \span
	\nonumber
\\ \Leftrightarrow& &
	- \bb\aa
	\frac{m}{\gcd(m,2n^2)}
	&\equiv k &&
	\label{eqn:better_eps_problem_k_mod_n}
\\ &&&
	\mod \gcd\left(
		\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
		\frac{mn \aa}{\gcd(m,2n^2)}
	\right)
	\span \span \span
	\nonumber
\end{align}

In our situation, we want to find the least $k>0$ satisfying
Equation \ref{eqn:finding_better_eps_problem}.
Since such a $k$ must also satisfy Equation \ref{eqn:better_eps_problem_k_mod_n},
we can pick the smallest $k_{q,v} \in \ZZ_{>0}$ which satisfies this new condition
(a computation only depending on $q$ and $\beta$, but not $r$).
We are then guaranteed that $k_{v,q}$ is less than any $k$ satisfying Equation
\ref{eqn:finding_better_eps_problem}, giving the first inequality in Equation
\ref{eqn:epsilon_q_lemma_prop}.
Furthermore, $k_{v,q}\geq 1$ gives the second part of the inequality:
$\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.

\end{proof}

\begin{sagesilent}
from plots_and_expressions import main_theorem2
\end{sagesilent}
\begin{theorem}[Third bound on $r$ for Problem \ref{problem:problem-statement-2}]
\label{thm:rmax_with_eps1}
	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.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},
	and $R = \chern_0(v)$
\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.

\begin{sagesilent}
from plots_and_expressions import main_theorem2_corollary
\end{sagesilent}
\begin{corollary}[Third bound on $r$ on $\PP^2$ and principally polarised abelian surfaces]
\label{cor:rmax_with_eps1}
	Suppose we are working over $\PP^2$ or a principally polarised abelian surface
	(or any other surfaces with $m=\ell^2=1$ or $2$).
	Let $v$ be a fixed Chern character, with $\beta_{-}\coloneqq\beta_{-}(v)=\frac{a_v}{n}$
	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:

	\begin{align*}
		\min
		\left(
			\sage{main_theorem2_corollary.r_upper_bound1.subs(betamin_subs)}, \:\:
			\sage{main_theorem2_corollary.r_upper_bound2.subs(betamin_subs)}
		\right),
	\end{align*}
	where $R = \chern_0(v)$ and $k_{v,q}$ is the least
	$k\in\ZZ_{>0}$ satisfying
	${
		k \equiv -\aa\bb
		\pmod{n}
	}$.
\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}
Just like in Examples \ref{exmpl:recurring-first} and
\ref{exmpl:recurring-second},
take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
$\beta=\sage{recurring.betaminus}$, giving $n=\sage{recurring.n}$
and $\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
%% TODO transcode notebook code
The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabiliser $u$ of $v$
in terms of the possible values for $q\coloneqq\chern_1^{\beta_{-}}(u)$ are as follows:

\begin{sagesilent}
from examples import bound_comparisons
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
\end{sagesilent}

\vspace{1em}
\noindent
\directlua{ table_width = 3*4+1 }
\begin{tabular}{l\directlua{for i=0,table_width-1 do tex.sprint([[|c]]) end}}
	$q=\chern_1^{\beta_{-}}(u)$
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	\\ \hline
	Theorem \ref{thm:rmax_with_uniform_eps}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	\\
	Theorem \ref{thm:rmax_with_eps1}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
\end{tabular}
\vspace{1em}

\noindent
It is worth noting that the bounds given by Theorem \ref{thm:rmax_with_eps1}
reach, but do not exceed, the actual maximum rank 25 of the
pseudo-semistabilisers of $v$ in this case.
As a reminder, the original loose bound from Theorem \ref{thm:loose-bound-on-r}
was 144.

\end{example}

\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-third}
Just like in examples \ref{exmpl:extravagant-first} and
\ref{exmpl:extravagant-second},
take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
$\beta=\sage{extravagant.betaminus}$, giving $n=\sage{n:=extravagant.n}$
and $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
This example was chosen because the $n$ value is moderatly large, giving more
possible values for $k_{v,q}$, in Definition/Lemma \ref{lemdfn:epsilon_q}. This allows
for a larger possible difference between the bounds given by Theorems
\ref{thm:rmax_with_uniform_eps} and \ref{thm:rmax_with_eps1}, with the bound
from the second being up to $\sage{n}$ times smaller, for any given $q$ value.
The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabiliser $u$ of $v$
in terms of the first few smallest possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:

\begin{sagesilent}
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(extravagant)
\end{sagesilent}


\vspace{1em}
\noindent
\directlua{ table_width = 12 }
\begin{tabular}{l\directlua{for i=0,table_width do tex.sprint([[|c]]) end}}
	$q=\chern_1^\beta(u)$
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	&$\cdots$
	\\ \hline
	Theorem \ref{thm:rmax_with_uniform_eps}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	&$\cdots$
	\\
	Theorem \ref{thm:rmax_with_eps1}
\directlua{for i=0,table_width-1 do
	local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
  tex.sprint(cell)
end}
	&$\cdots$
\end{tabular}
\vspace{1em}


\noindent
However the reduction in the overall bound on $r$ is not as drastic, since all
possible values for $k_{v,q}$ in $\{1,2,\ldots,\sage{n}\}$ are iterated through
cyclically as we consider successive possible values for $q$.
And for each $q$ where $k_{v,q}=1$, both theorems give the same bound.
Calculating the maximums over all values of $q$ yields
$\sage{max(theorem2_bounds)}$ for Theorem \ref{thm:rmax_with_uniform_eps}, and
$\sage{max(theorem3_bounds)}$ for Theorem \ref{thm:rmax_with_eps1}.
\end{example}