bounds-on-semistabilisers.tex

	\noindent
	And $r$ satisfies $\aa r + \bb \equiv 0 \pmod{n}$,
	then we have:

	\begin{equation}
		\label{eqn:epsilon_q_lemma_prop}
		d - \frac{(\aa r + 2\bb)\aa}{2n^2}
		\geq \epsilon_{v,q} \geq \epsilon_v > 0
	\end{equation}

	\noindent
	Where $\epsilon_{v,q}$ is defined as follows:

	\begin{equation*}
		\epsilon_{v,q} \coloneqq
		\frac{k_{q}}{\lcm(m,2n^2)}
	\end{equation*}
	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 \eqref{eqn:finding_better_eps_problem}.
	Since such a $k$ must also satisfy Equation \eqref{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
	\eqref{eqn:finding_better_eps_problem}, giving the first inequality in Equation
	\eqref{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
	with $\mathrm{Pic}(\ppas) = \ZZ\ell$
	(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}