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tex/bounds-on-semistabilisers.tex
View file @ 1a27cd54
...@@ -4,6 +4,7 @@ ...@@ -4,6 +4,7 @@
The proof for the following Theorem \ref{thm:loose-bound-on-r} was hinted at in The proof for the following Theorem \ref{thm:loose-bound-on-r} was hinted at in
\cite{SchmidtBenjamin2020Bsot}, but the value appears explicitly in \cite{SchmidtBenjamin2020Bsot}, but the value appears explicitly in
\cite{SchmidtGithub2020} as shown in the following Listing \cite{SchmidtGithub2020} as shown in the following Listing
% texlab: ignore
\ref{fig:code:schmidt-bound}. \ref{fig:code:schmidt-bound}.
The latter citation is a SageMath \cite{sagemath} The latter citation is a SageMath \cite{sagemath}
library for computing certain quantities related to Bridgeland stabilities on library for computing certain quantities related to Bridgeland stabilities on
...@@ -18,130 +19,130 @@ pseudo-semistabilisers for tilt stability. ...@@ -18,130 +19,130 @@ pseudo-semistabilisers for tilt stability.
]{schmidt-snippet} ]{schmidt-snippet}
\begin{theorem}[Bound on $r$ - Benjamin Schmidt] \begin{theorem}[Bound on $r$ - Benjamin Schmidt]
\label{thm:loose-bound-on-r} \label{thm:loose-bound-on-r}
Let $X$ be a smooth projective Picard rank 1 surface with choice of ample line Let $X$ be a smooth projective Picard rank 1 surface with choice of ample line
bundle $L$ such that $\ell\coloneqq c_1(L)$ generates $\neronseveri(X)$ and bundle $L$ such that $\ell\coloneqq c_1(L)$ generates $\neronseveri(X)$ and
take $m\coloneqq \ell^2$. take $m\coloneqq \ell^2$.
Given a Chern character $v$ with $\Delta(v) \geq 0$ and positive rank (or Given a Chern character $v$ with $\Delta(v) \geq 0$ and positive rank (or
$\chern_0(v) = 0$ and $\chern_1(v) > 0$) $\chern_0(v) = 0$ and $\chern_1(v) > 0$)
such that such that
$\beta_-\coloneqq\beta_{-}(v)\in\QQ$, the rank $r$ of $\beta_-\coloneqq\beta_{-}(v)\in\QQ$, the rank $r$ of
any solution $u$ of Problem \ref{problem:problem-statement-2} is any solution $u$ of Problem \ref{problem:problem-statement-2} is
bounded above by: bounded above by:
\begin{equation*} \begin{equation*}
r \leq \frac{m\left(n \chern^{\beta_-}_1(v) - 1\right)^2}{\gcd(m,2n^2)} r \leq \frac{m\left(n \chern^{\beta_-}_1(v) - 1\right)^2}{\gcd(m,2n^2)}
\end{equation*} \end{equation*}
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}
The Bogomolov form applied to the twisted Chern character is the same as the The Bogomolov form applied to the twisted Chern character is the same as the
untwisted one. untwisted one.
\noindent
\begin{minipage}{0.57\linewidth}
So $0 \leq \Delta(u)$ (condition 2 from Corollary \ref{cor:num_test_prob2})
yields:
\begin{equation}
\label{eqn-bgmlv-on-E}
2\chern_0(u) \chern^{\beta_{-}}_2(u) \leq \chern^{\beta_{-}}_1(u)^2
\end{equation}
\noindent \noindent
Furthermore, \begin{minipage}{0.57\linewidth}
condition 5 from Corollary \ref{cor:num_test_prob2} So $0 \leq \Delta(u)$ (condition 2 from Corollary \ref{cor:num_test_prob2})
gives: yields:
\begin{equation} \begin{equation}
\label{eqn-tilt-cat-cond} \label{eqn-bgmlv-on-E}
0 < \chern^{\beta_{-}}_1(u) < \chern^{\beta_{-}}_1(v) 2\chern_0(u) \chern^{\beta_{-}}_2(u) \leq \chern^{\beta_{-}}_1(u)^2
\end{equation} \end{equation}
\noindent
Furthermore,
condition 5 from Corollary \ref{cor:num_test_prob2}
gives:
\begin{equation}
\label{eqn-tilt-cat-cond}
0 < \chern^{\beta_{-}}_1(u) < \chern^{\beta_{-}}_1(v)
\end{equation}
\noindent
The induced restrictions on possible pairs $\chern^{\beta_-}_0(u)$ and
$\chern^{\beta_-}_2(u)$,
as well as conditions 1 and 6 from Corollary \ref{cor:num_test_prob2}
are illustrated here on the right, with the invalid regions shaded.
\end{minipage}
\hfill
\begin{minipage}{0.39\linewidth}
%\label{prop:proof:fig:pseudowall-pos}
\begin{center}
\def\svgwidth{\linewidth}
{\small
\subimport{../figures/}{schmidt-arg-diag.pdf_tex}
}
\end{center}
\vspace{3pt}
\end{minipage}
Currently, the unshaded region in the diagram above, corresponding to possible
values for $\chern_0(u)$ and $\chern^{\beta_{-}}_2(u)$ that satisfy the
currently considered restrictions, is unbounded.
This is where the rationality of $\beta_{-}$ comes in.
If $\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$,
then $\chern^{\beta_-}_2(u) \in \frac{1}{\lcm(m,2n^2)}\ZZ$.
In particular, since $\chern_2^{\beta_-}(u) > 0$ we must also have
$\chern^{\beta_-}_2(u) \geq \frac{1}{\lcm(m,2n^2)}$, which then in turn gives a
bound for the rank of $u$:
\begin{align}
\chern_0(u)
& \leq \frac{\chern^{\beta_-}_1(u)^2}{2\chern^{\beta_{-}}_2(u)} \nonumber \\
& \leq \frac{\lcm(m,2n^2) \chern^{\beta_-}_1(u)^2}{2} \nonumber \\
& = \frac{mn^2 \chern^{\beta_-}_1(u)^2}{\gcd(m,2n^2)}
\label{proof:first-bound-on-r}
\end{align}
\noindent \noindent
The induced restrictions on possible pairs $\chern^{\beta_-}_0(u)$ and Which we can then immediately bound using Equation \ref{eqn-tilt-cat-cond}.
$\chern^{\beta_-}_2(u)$, Alternatively, given that
as well as conditions 1 and 6 from Corollary \ref{cor:num_test_prob2} $\chern_1^{\beta_{-}}(u)$, $\chern_1^{\beta_{-}}(v)\in\frac{1}{n}\ZZ$,
are illustrated here on the right, with the invalid regions shaded. we can tighten this bound on $\chern_1^{\beta_{-}}(u)$ given by that equation to:
\end{minipage} \[
\hfill n\chern^{\beta_{-}}_1(u) \leq n\chern^{\beta_{-}}_1(v) - 1
\begin{minipage}{0.39\linewidth} \]
%\label{prop:proof:fig:pseudowall-pos} allowing us to bound the expression in Equation \ref{proof:first-bound-on-r} to
\begin{center} the following:
\def\svgwidth{\linewidth} \[
{\small \chern_0(u)
\subimport{../figures/}{schmidt-arg-diag.pdf_tex} \leq \frac{m\left(n \chern^{\beta_-}_1(v) - 1\right)^2}{\gcd(m,2n^2)}
} \]
\end{center}
\vspace{3pt}
\end{minipage}
Currently, the unshaded region in the diagram above, corresponding to possible
values for $\chern_0(u)$ and $\chern^{\beta_{-}}_2(u)$ that satisfy the
currently considered restrictions, is unbounded.
This is where the rationality of $\beta_{-}$ comes in.
If $\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$,
then $\chern^{\beta_-}_2(u) \in \frac{1}{\lcm(m,2n^2)}\ZZ$.
In particular, since $\chern_2^{\beta_-}(u) > 0$ we must also have
$\chern^{\beta_-}_2(u) \geq \frac{1}{\lcm(m,2n^2)}$, which then in turn gives a
bound for the rank of $u$:
\begin{align}
\chern_0(u)
&\leq \frac{\chern^{\beta_-}_1(u)^2}{2\chern^{\beta_{-}}_2(u)} \nonumber \\
&\leq \frac{\lcm(m,2n^2) \chern^{\beta_-}_1(u)^2}{2} \nonumber \\
&= \frac{mn^2 \chern^{\beta_-}_1(u)^2}{\gcd(m,2n^2)}
\label{proof:first-bound-on-r}
\end{align}
\noindent
Which we can then immediately bound using Equation \ref{eqn-tilt-cat-cond}.
Alternatively, given that
$\chern_1^{\beta_{-}}(u)$, $\chern_1^{\beta_{-}}(v)\in\frac{1}{n}\ZZ$,
we can tighten this bound on $\chern_1^{\beta_{-}}(u)$ given by that equation to:
\[
n\chern^{\beta_{-}}_1(u) \leq n\chern^{\beta_{-}}_1(v) - 1
\]
allowing us to bound the expression in Equation \ref{proof:first-bound-on-r} to
the following:
\[
\chern_0(u)
\leq \frac{m\left(n \chern^{\beta_-}_1(v) - 1\right)^2}{\gcd(m,2n^2)}
\]
\end{proof} \end{proof}
\begin{sagesilent} \begin{sagesilent}
from examples import recurring from examples import recurring
\end{sagesilent} \end{sagesilent}
\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$] \begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-first} \label{exmpl:recurring-first}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=1$, $\beta_-=\sage{recurring.betaminus}$, that $m=1$, $\beta_-=\sage{recurring.betaminus}$,
giving $n=\sage{recurring.n}$ and giving $n=\sage{recurring.n}$ and
$\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$. $\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
tilt semistabilisers for $v$ are bounded above by $\sage{recurring.loose_bound}$. tilt semistabilisers for $v$ are bounded above by $\sage{recurring.loose_bound}$.
However, when computing all tilt semistabilisers for $v$ on $\PP^2$, the maximum However, when computing all tilt semistabilisers for $v$ on $\PP^2$, the maximum
rank that appears turns out to be 25. This will be a recurring example to rank that appears turns out to be 25. This will be a recurring example to
illustrate the performance of later theorems about rank bounds illustrate the performance of later theorems about rank bounds
\end{example} \end{example}
\begin{sagesilent} \begin{sagesilent}
from examples import extravagant from examples import extravagant
\end{sagesilent} \end{sagesilent}
\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$] \begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-first} \label{exmpl:extravagant-first}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=1$, $\beta_-=\sage{extravagant.betaminus}$, that $m=1$, $\beta_-=\sage{extravagant.betaminus}$,
giving $n=\sage{extravagant.n}$ and giving $n=\sage{extravagant.n}$ and
$\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$. $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
tilt semistabilisers for $v$ are bounded above by $\sage{extravagant.loose_bound}$. tilt semistabilisers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
However, when computing all tilt semistabilisers for $v$ on $\PP^2$, the maximum However, when computing all tilt semistabilisers for $v$ on $\PP^2$, the maximum
rank that appears turns out to be $\sage{round(extravagant.actual_rmax, 1)}$. rank that appears turns out to be $\sage{round(extravagant.actual_rmax, 1)}$.
\end{example} \end{example}
...@@ -166,19 +167,19 @@ and Corollary \ref{cor:num_test_prob2}, in a way which better fits our direction ...@@ -166,19 +167,19 @@ and Corollary \ref{cor:num_test_prob2}, in a way which better fits our direction
of travel. of travel.
\begin{lemma} \begin{lemma}
\label{lem:fixed-q-semistabs-criterion} \label{lem:fixed-q-semistabs-criterion}
Consider the Problem \ref{problem:problem-statement-1} (or \ref{problem:problem-statement-2}), Consider the Problem \ref{problem:problem-statement-1} (or \ref{problem:problem-statement-2}),
with choice of point $P=(\alpha_0,\beta_0)$ on $\Theta_v^{-}$ with choice of point $P=(\alpha_0,\beta_0)$ on $\Theta_v^{-}$
(or $\alpha_0=0$ and $\beta_0=\beta_{-}(v)$ resp.). (or $\alpha_0=0$ and $\beta_0=\beta_{-}(v)$ resp.).
\noindent \noindent
If $u$ is a solution to the Problem then $u$ satisfies: If $u$ is a solution to the Problem then $u$ satisfies:
\begin{equation} \begin{equation}
q\coloneqq \chern^{\beta_0}_1(u) \in \left( 0, \chern_1^{\beta_0}(v) \right) q\coloneqq \chern^{\beta_0}_1(u) \in \left( 0, \chern_1^{\beta_0}(v) \right)
\label{lem:eqn:cond-for-fixed-q} \label{lem:eqn:cond-for-fixed-q}
\qquad \qquad
\text{and} \text{and}
\qquad \qquad
\chern_0(u) > \frac{q}{\mu(v) - \beta_0}. \chern_0(u) > \frac{q}{\mu(v) - \beta_0}.
\nonumber \nonumber
\end{equation} \end{equation}
...@@ -189,11 +190,11 @@ of travel. ...@@ -189,11 +190,11 @@ of travel.
satisfying the above Equations \ref{lem:eqn:cond-for-fixed-q} satisfying the above Equations \ref{lem:eqn:cond-for-fixed-q}
is a solution to the Problem if and only if the following are satisfied: is a solution to the Problem if and only if the following are satisfied:
\begin{multicols}{3} \begin{multicols}{3}
\begin{itemize} \begin{itemize}
\item $\Delta(u) \geq 0$ \item $\Delta(u) \geq 0$
\item $\Delta(v-u) \geq 0$ \item $\Delta(v-u) \geq 0$
\item $\chern^{\alpha_0,\beta_0}_2(u) \geq 0$ \item $\chern^{\alpha_0,\beta_0}_2(u) \geq 0$
\end{itemize} \end{itemize}
\end{multicols} \end{multicols}
\end{lemma} \end{lemma}
...@@ -202,23 +203,23 @@ of travel. ...@@ -202,23 +203,23 @@ of travel.
to the problem are given by $u=(r,c\ell,d\ell^2)$ with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$ to the problem are given by $u=(r,c\ell,d\ell^2)$ with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$
which satisfy six numerical conditions. which satisfy six numerical conditions.
The first line of Equation \ref{lem:eqn:cond-for-fixed-q} is equivalent to The first line of Equation \ref{lem:eqn:cond-for-fixed-q} is equivalent to
numerical condition 5. numerical condition 5.
The second line is a rearrangement of numerical condition 4, assuming $r>0$ which is given by The second line is a rearrangement of numerical condition 4, assuming $r>0$ which is given by
the first numerical condition. the first numerical condition.
Therefore any solution $u$ satisfies Equation \ref{lem:eqn:cond-for-fixed-q}. Therefore any solution $u$ satisfies Equation \ref{lem:eqn:cond-for-fixed-q}.
But then Theorems \ref{lem:num_test_prob1} and \ref{cor:num_test_prob2}, also give that But then Theorems \ref{lem:num_test_prob1} and \ref{cor:num_test_prob2}, also give that
$u=(r,c\ell,d\ell^2)$ with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$ satisfying Equation $u=(r,c\ell,d\ell^2)$ with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$ satisfying Equation
\ref{lem:eqn:cond-for-fixed-q} is in fact a solution provided the remaining numerical conditions \ref{lem:eqn:cond-for-fixed-q} is in fact a solution provided the remaining numerical conditions
1, 2, 3 and 6 are satisfied. 1, 2, 3 and 6 are satisfied.
This is in essence the second part of the Lemma. This is in essence the second part of the Lemma.
\end{proof} \end{proof}
\begin{corollary} \begin{corollary}
\label{cor:rational-beta:fixed-q-semistabs-criterion} \label{cor:rational-beta:fixed-q-semistabs-criterion}
Consider the Problem \ref{problem:problem-statement-1} (or \ref{problem:problem-statement-2}), Consider the Problem \ref{problem:problem-statement-1} (or \ref{problem:problem-statement-2}),
with choice of point $P=(\alpha_0,\beta_0)$ on $\Theta_v^{-}$ with choice of point $P=(\alpha_0,\beta_0)$ on $\Theta_v^{-}$
(or $\alpha_0=0$ and $\beta_0=\beta_{-}(v)$ resp.), (or $\alpha_0=0$ and $\beta_0=\beta_{-}(v)$ resp.),
and suppose that $\beta_{0}$ is and suppose that $\beta_{0}$ is
rational, and written $\beta_0=\frac{a_v}{n}$ for rational, and written $\beta_0=\frac{a_v}{n}$ for
some coprime integers $a_v$, $n$ with $n>0$. some coprime integers $a_v$, $n$ with $n>0$.
...@@ -227,27 +228,27 @@ of travel. ...@@ -227,27 +228,27 @@ of travel.
\begin{align*} \begin{align*}
\chern^{\beta_0}_1(u) \chern^{\beta_0}_1(u)
= \frac{b_q}{n}, = \frac{b_q}{n},
\qquad \qquad
a_v r &\equiv -b_q \pmod{n}, a_v r & \equiv -b_q \pmod{n},
\quad \quad
\text{and} \text{and}
\qquad \qquad
r > \frac{q}{\mu(v) - \beta_0} r > \frac{q}{\mu(v) - \beta_0}
\end{align*} \end{align*}
\[ \[
\text{for some } \text{for some }
b_q \in \left\{ 1, 2, \ldots, n\chern_1^{\beta_0}(v) - 1 \right\}. b_q \in \left\{ 1, 2, \ldots, n\chern_1^{\beta_0}(v) - 1 \right\}.
\] \]
And any $u = (r,c\ell,d\ell^2)$ And any $u = (r,c\ell,d\ell^2)$
with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$ with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$
satisfying these equations is a solution to the Problem if and only if, again, satisfying these equations is a solution to the Problem if and only if, again,
the following are satisfied: the following are satisfied:
\begin{multicols}{3} \begin{multicols}{3}
\begin{itemize} \begin{itemize}
\item $\Delta(u) \geq 0$ \item $\Delta(u) \geq 0$
\item $\Delta(v-u) \geq 0$ \item $\Delta(v-u) \geq 0$
\item $\chern^P_2(u) \geq 0$ \item $\chern^P_2(u) \geq 0$
\end{itemize} \end{itemize}
\end{multicols} \end{multicols}
\end{corollary} \end{corollary}
...@@ -256,24 +257,24 @@ of travel. ...@@ -256,24 +257,24 @@ of travel.
This is a specialisation of Lemma \ref{lem:fixed-q-semistabs-criterion} This is a specialisation of Lemma \ref{lem:fixed-q-semistabs-criterion}
with a modification to the statement with a modification to the statement
\[ \[
q\coloneqq \chern^{\beta_0}_1(u) \in \left( 0, \chern_1^{\beta_0}(v) \right) q\coloneqq \chern^{\beta_0}_1(u) \in \left( 0, \chern_1^{\beta_0}(v) \right)
\] \]
for the case where $\beta_0$ is rational. for the case where $\beta_0$ is rational.
Taking $\beta_0 = \frac{a_v}{n}$ we have: Taking $\beta_0 = \frac{a_v}{n}$ we have:
\[ \[
q\coloneqq\chern_1^{\beta_0}(u) q\coloneqq\chern_1^{\beta_0}(u)
= c - \frac{a_v}{n}r = c - \frac{a_v}{n}r
\in \frac{1}{n}\ZZ \in \frac{1}{n}\ZZ
\] \]
So $q=\frac{b_q}{n}$ for some $b_q \in \left\{ 1, 2, \ldots, n\chern_1^{\beta_0}(v) - 1 \right\}$ So $q=\frac{b_q}{n}$ for some $b_q \in \left\{ 1, 2, \ldots, n\chern_1^{\beta_0}(v) - 1 \right\}$
and then and then
${ ${
nc - a_v r = b_q nc - a_v r = b_q
}$ }$
and so and so
${ ${
a_v r \equiv -b_q a_v r \equiv -b_q
}$ modulo $n$. }$ modulo $n$.
\end{proof} \end{proof}
\subsection{Bounds on \texorpdfstring{`$d$'}{d}-values for Solutions of Problems} \subsection{Bounds on \texorpdfstring{`$d$'}{d}-values for Solutions of Problems}
...@@ -299,7 +300,7 @@ to the Problem satisfies ...@@ -299,7 +300,7 @@ to the Problem satisfies
q \coloneqq \chern_1^{\beta_0}(u) q \coloneqq \chern_1^{\beta_0}(u)
\in \in
\left( \left(
0, \chern_1^{\beta_0}(v) 0, \chern_1^{\beta_0}(v)
\right) \right)
\] \]
and also gives a lower bound for $r$ when considering $u$ with a fixed $q$. and also gives a lower bound for $r$ when considering $u$ with a fixed $q$.
...@@ -322,7 +323,7 @@ In the context of Problem \ref{problem:problem-statement-2}, this condition, ...@@ -322,7 +323,7 @@ In the context of Problem \ref{problem:problem-statement-2}, this condition,
when rearranged to a bound on $d$, amounts to: when rearranged to a bound on $d$, amounts to:
\begin{equation} \begin{equation}
\label{eqn:radius-cond-betamin} \label{eqn:radius-cond-betamin}
\chern_2^{\beta_{-}}(u) > 0 \chern_2^{\beta_{-}}(u) > 0
\qquad \qquad
\text{and} \text{and}
...@@ -332,7 +333,7 @@ when rearranged to a bound on $d$, amounts to: ...@@ -332,7 +333,7 @@ when rearranged to a bound on $d$, amounts to:
\end{equation} \end{equation}
\begin{sagesilent} \begin{sagesilent}
import other_P_choice as problem1 import other_P_choice as problem1
\end{sagesilent} \end{sagesilent}
In the case where we are tackling Problem \ref{problem:problem-statement-1}, In the case where we are tackling Problem \ref{problem:problem-statement-1},
...@@ -368,7 +369,7 @@ $q = \chern^{\beta_0}_1(u) = c - r\beta_0$, ...@@ -368,7 +369,7 @@ $q = \chern^{\beta_0}_1(u) = c - r\beta_0$,
we get: we get:
\begin{sagesilent} \begin{sagesilent}
from plots_and_expressions import bgmlv2_with_q from plots_and_expressions import bgmlv2_with_q
\end{sagesilent} \end{sagesilent}
\begin{equation} \begin{equation}
\sage{bgmlv2_with_q} \sage{bgmlv2_with_q}
...@@ -379,7 +380,7 @@ Rearranging to express this as a bound on $d$, we get the following. ...@@ -379,7 +380,7 @@ Rearranging to express this as a bound on $d$, we get the following.
Recall that $r>0$ is ensured by Equations \ref{lem:eqn:cond-for-fixed-q}. Recall that $r>0$ is ensured by Equations \ref{lem:eqn:cond-for-fixed-q}.
\begin{sagesilent} \begin{sagesilent}
from plots_and_expressions import bgmlv2_d_ineq from plots_and_expressions import bgmlv2_d_ineq
\end{sagesilent} \end{sagesilent}
\begin{equation} \begin{equation}
\label{eqn-bgmlv2_d_upperbound} \label{eqn-bgmlv2_d_upperbound}
...@@ -400,7 +401,7 @@ $d$ yields: ...@@ -400,7 +401,7 @@ $d$ yields:
\begin{sagesilent} \begin{sagesilent}
from plots_and_expressions import bgmlv3_d_upperbound_terms from plots_and_expressions import bgmlv3_d_upperbound_terms
\end{sagesilent} \end{sagesilent}
\begin{equation} \begin{equation}
...@@ -433,31 +434,31 @@ see-saw principle. ...@@ -433,31 +434,31 @@ see-saw principle.
% TODO maybe cover the see-saw principle % TODO maybe cover the see-saw principle
\begin{align*} \begin{align*}
\left( \left(
\frac{\chern^{\beta_0}_1(v-u)}{\chern_0(v-u)} \frac{\chern^{\beta_0}_1(v-u)}{\chern_0(v-u)}
\right)^2 \right)^2
&= & =
\left( \left(
\mu(v-u) - \beta_0 \mu(v-u) - \beta_0
\right)^2 \right)^2
\\ \\
&> & >
\left( \left(
\mu(v) - \beta_0 \mu(v) - \beta_0
\right)^2 \right)^2
&\text{by Equation \ref{lem:proof:slope-order-rltR}} & \text{by Equation \ref{lem:proof:slope-order-rltR}}
\\ \\
&= & =
\left( \left(
\frac{\chern^{\beta_0}_1(v)}{\chern_0(v)} \frac{\chern^{\beta_0}_1(v)}{\chern_0(v)}
\right)^2 \right)^2
\\ \\
&\geq & \geq
2 \frac{\chern^{\beta_0}_2(v)}{\chern_0(v)} 2 \frac{\chern^{\beta_0}_2(v)}{\chern_0(v)}
&\text{since }\Delta(v) \geq 0 & \text{since }\Delta(v) \geq 0
\:\text{and }\chern_0(v) > 0 \:\text{and }\chern_0(v) > 0
\\ \\
\text{So} \text{So}
\quad \quad
\frac{ \frac{
\left( \left(
q-\chern^{\beta_0}_1(v) q-\chern^{\beta_0}_1(v)
...@@ -467,9 +468,9 @@ see-saw principle. ...@@ -467,9 +468,9 @@ see-saw principle.
R-r R-r
\right)^2 \right)^2
} }
&> & >
2 \frac{\chern^{\beta_0}_2(v)}{R} 2 \frac{\chern^{\beta_0}_2(v)}{R}
& &
\text{and} \text{and}
\quad \quad
\chern_2^{\beta_0}(v) \chern_2^{\beta_0}(v)
...@@ -482,7 +483,7 @@ see-saw principle. ...@@ -482,7 +483,7 @@ see-saw principle.
R-r R-r
\right) \right)
} }
&< & <
\frac{r\chern^{\beta_0}_2(v)}{R} \frac{r\chern^{\beta_0}_2(v)}{R}
\end{align*} \end{align*}
\noindent \noindent
...@@ -493,7 +494,7 @@ are greater than those of Equation ...@@ -493,7 +494,7 @@ are greater than those of Equation
\subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem \subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
\texorpdfstring{\ref{problem:problem-statement-2}}{2}} \texorpdfstring{\ref{problem:problem-statement-2}}{2}}
\label{subsubsect:all-bounds-on-d-prob2} \label{subsubsect:all-bounds-on-d-prob2}
In the context of Problem \ref{problem:problem-statement-2}, with In the context of Problem \ref{problem:problem-statement-2}, with
...@@ -505,30 +506,30 @@ for a potential solution to the problem of the form in Equation ...@@ -505,30 +506,30 @@ for a potential solution to the problem of the form in Equation
\ref{eqn:u-coords}, amounts to the following: \ref{eqn:u-coords}, amounts to the following:
\begin{sagesilent} \begin{sagesilent}
from plots_and_expressions import bgmlv2_d_upperbound_terms from plots_and_expressions import bgmlv2_d_upperbound_terms
\end{sagesilent} \end{sagesilent}
\begin{align} \begin{align}
d &> d & >
\frac{1}{2}{\beta_0}^2 r \frac{1}{2}{\beta_0}^2 r
+ {\beta_0} q, + {\beta_0} q,
\phantom{+} % to keep terms aligned \phantom{+} % to keep terms aligned
&\qquad\text{when\:} r > 0 & \qquad\text{when\:} r > 0
\label{eqn:radiuscond_d_bound_betamin} \label{eqn:radiuscond_d_bound_betamin}
\\ \\
d &\leq d & \leq
\sage{bgmlv2_d_upperbound_terms.problem2.linear} \sage{bgmlv2_d_upperbound_terms.problem2.linear}
+ \sage{bgmlv2_d_upperbound_terms.problem2.const} + \sage{bgmlv2_d_upperbound_terms.problem2.const}
+\sage{bgmlv2_d_upperbound_terms.problem2.hyperbolic}, +\sage{bgmlv2_d_upperbound_terms.problem2.hyperbolic},
&\qquad\text{when\:} r > 0 & \qquad\text{when\:} r > 0
\label{eqn:bgmlv2_d_bound_betamin} \label{eqn:bgmlv2_d_bound_betamin}
\\ \\
d &\leq d & \leq
\sage{bgmlv3_d_upperbound_terms.problem2.linear} \sage{bgmlv3_d_upperbound_terms.problem2.linear}
+ \sage{bgmlv3_d_upperbound_terms.problem2.const} + \sage{bgmlv3_d_upperbound_terms.problem2.const}
% ^ ch_2^\beta(F)=0 for beta_{-} % ^ ch_2^\beta(F)=0 for beta_{-}
\sage{bgmlv3_d_upperbound_terms.problem2.hyperbolic}, \sage{bgmlv3_d_upperbound_terms.problem2.hyperbolic},
&\qquad\text{when\:} r > R & \qquad\text{when\:} r > R
\label{eqn:bgmlv3_d_bound_betamin} \label{eqn:bgmlv3_d_bound_betamin}
\end{align} \end{align}
Recalling that $q \coloneqq \chern^{\beta}_1(u) \in (0, \chern^{\beta}_1(v))$, Recalling that $q \coloneqq \chern^{\beta}_1(u) \in (0, \chern^{\beta}_1(v))$,
...@@ -560,22 +561,22 @@ This will be pursued in Subsection ...@@ -560,22 +561,22 @@ This will be pursued in Subsection
\ref{subsec:bounds-on-semistab-rank-prob-2}. \ref{subsec:bounds-on-semistab-rank-prob-2}.
\begin{sagesilent} \begin{sagesilent}
from plots_and_expressions import typical_bounds_on_d from plots_and_expressions import typical_bounds_on_d
\end{sagesilent} \end{sagesilent}
\begin{figure} \begin{figure}
\centering \centering
\sageplot[width=\linewidth]{typical_bounds_on_d} \sageplot[width=\linewidth]{typical_bounds_on_d}
\caption{ \caption{
Bounds on $d\coloneqq\chern_2(u)$ in terms of $r\coloneqq \chern_0(u)$ for a fixed Bounds on $d\coloneqq\chern_2(u)$ in terms of $r\coloneqq \chern_0(u)$ for a fixed
value $\chern_1^{\beta}(v)/2$ of $q\coloneqq\chern_1^{\beta}(u)$. value $\chern_1^{\beta}(v)/2$ of $q\coloneqq\chern_1^{\beta}(u)$.
Where $\chern(v) = (3,2\ell,-2\ell^2)$. Where $\chern(v) = (3,2\ell,-2\ell^2)$.
} }
\label{fig:d_bounds_xmpl_gnrc_q} \label{fig:d_bounds_xmpl_gnrc_q}
\end{figure} \end{figure}
\subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem \subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
\texorpdfstring{\ref{problem:problem-statement-1}}{1}} \texorpdfstring{\ref{problem:problem-statement-1}}{1}}
\label{subsubsect:all-bounds-on-d-prob1} \label{subsubsect:all-bounds-on-d-prob1}
Unlike for Problem \ref{problem:problem-statement-2}, Unlike for Problem \ref{problem:problem-statement-2},
...@@ -588,37 +589,37 @@ bounds do not share the same assymptote as the lower bound ...@@ -588,37 +589,37 @@ bounds do not share the same assymptote as the lower bound
\begin{align} \begin{align}
\sage{problem1.radius_condition_d_bound.lhs()} \sage{problem1.radius_condition_d_bound.lhs()}
&> & >
\sage{problem1.radius_condition_d_bound.rhs()} \sage{problem1.radius_condition_d_bound.rhs()}
&\text{when }r>0 & \text{when }r>0
\label{eqn:prob1:radiuscond} \label{eqn:prob1:radiuscond}
\\ \\
d &\leq d & \leq
\sage{problem1.bgmlv2_d_upperbound_terms.linear} \sage{problem1.bgmlv2_d_upperbound_terms.linear}
+ \sage{problem1.bgmlv2_d_upperbound_terms.const} + \sage{problem1.bgmlv2_d_upperbound_terms.const}
+ \sage{problem1.bgmlv2_d_upperbound_terms.hyperbolic} + \sage{problem1.bgmlv2_d_upperbound_terms.hyperbolic}
&\text{when }r>R & \text{when }r>R
\label{eqn:prob1:bgmlv2} \label{eqn:prob1:bgmlv2}
\\ \\
d &\leq d & \leq
\sage{problem1.bgmlv3_d_upperbound_terms.linear} \sage{problem1.bgmlv3_d_upperbound_terms.linear}
+ \sage{problem1.bgmlv3_d_upperbound_terms.const} + \sage{problem1.bgmlv3_d_upperbound_terms.const}
\sage{problem1.bgmlv3_d_upperbound_terms.hyperbolic} \sage{problem1.bgmlv3_d_upperbound_terms.hyperbolic}
&\text{when }r>R & \text{when }r>R
\label{eqn:prob1:bgmlv3} \label{eqn:prob1:bgmlv3}
\end{align} \end{align}
\begin{figure} \begin{figure}
\centering \centering
\sageplot[width=\linewidth]{problem1.example_plot} \sageplot[width=\linewidth]{problem1.example_plot}
\caption{ \caption{
Bounds on $d\coloneqq\chern_2(u)$ in terms of $r\coloneqq \chern_0(u)$ for a fixed Bounds on $d\coloneqq\chern_2(u)$ in terms of $r\coloneqq \chern_0(u)$ for a fixed
value $\chern_1^{\beta}(v)/2$ of $q\coloneqq\chern_1^{\beta}(E)$. value $\chern_1^{\beta}(v)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
Where $\chern(v) = (3,2\ell,-2\ell^2)$ and $P$ chosen as the point on $\Theta_v$ Where $\chern(v) = (3,2\ell,-2\ell^2)$ and $P$ chosen as the point on $\Theta_v$
with $\beta(P)\coloneqq-2/3-1/99$ in the context of Problem with $\beta(P)\coloneqq-2/3-1/99$ in the context of Problem
\ref{problem:problem-statement-1}. \ref{problem:problem-statement-1}.
} }
\label{fig:problem1:d_bounds_xmpl_gnrc_q} \label{fig:problem1:d_bounds_xmpl_gnrc_q}
\end{figure} \end{figure}
...@@ -630,11 +631,11 @@ This is because $R\coloneqq\chern_0(v)$ ...@@ -630,11 +631,11 @@ This is because $R\coloneqq\chern_0(v)$
and $\chern_2^{\beta_0}(v)$ are all strictly positive: and $\chern_2^{\beta_0}(v)$ are all strictly positive:
\begin{itemize} \begin{itemize}
\item $R > 0$ from the setting of Problem \item $R > 0$ from the setting of Problem
\ref{problem:problem-statement-1} \ref{problem:problem-statement-1}
\item $\chern_2^{\beta_0}(v)>0$ \item $\chern_2^{\beta_0}(v)>0$
by Lemma \ref{lem:comparison-test-with-beta_} by Lemma \ref{lem:comparison-test-with-beta_}
because ${\beta_0} < \beta_{-}$ due to the choice of $P$ being because ${\beta_0} < \beta_{-}$ due to the choice of $P$ being
a point on $\Theta_v^{-}$ a point on $\Theta_v^{-}$
\end{itemize} \end{itemize}
This means that the lower bound for $d$ will be larger than either of the two This means that the lower bound for $d$ will be larger than either of the two
...@@ -645,7 +646,7 @@ A generic example of this is plotted in Figure ...@@ -645,7 +646,7 @@ A generic example of this is plotted in Figure
idea will be pursued in Subsection \ref{subsec:bounds-on-semistab-rank-prob-1}. idea will be pursued in Subsection \ref{subsec:bounds-on-semistab-rank-prob-1}.
\subsection{Bounds on Semistabiliser Rank \texorpdfstring{$r$}{} in Problem \subsection{Bounds on Semistabiliser Rank \texorpdfstring{$r$}{} in Problem
\ref{problem:problem-statement-1}} \ref{problem:problem-statement-1}}
\label{subsec:bounds-on-semistab-rank-prob-1} \label{subsec:bounds-on-semistab-rank-prob-1}
As discussed at the end of Subsection \ref{subsubsect:all-bounds-on-d-prob1} As discussed at the end of Subsection \ref{subsubsect:all-bounds-on-d-prob1}
...@@ -659,7 +660,7 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$ ...@@ -659,7 +660,7 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
\ref{fig:problem1:d_bounds_xmpl_gnrc_q}). \ref{fig:problem1:d_bounds_xmpl_gnrc_q}).
\begin{theorem}[Problem \ref{problem:problem-statement-1} upper Bound on $r$] \begin{theorem}[Problem \ref{problem:problem-statement-1} upper Bound on $r$]
\label{lem:prob1:r_bound} \label{lem:prob1:r_bound}
Let $u$ be a solution to Problem \ref{problem:problem-statement-1} Let $u$ be a solution to Problem \ref{problem:problem-statement-1}
and $q\coloneqq\chern_1^{B}(u)$. and $q\coloneqq\chern_1^{B}(u)$.
Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression: Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression:
...@@ -682,13 +683,13 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$ ...@@ -682,13 +683,13 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
Solving for the lower bound in Equation \ref{eqn:prob1:radiuscond} being Solving for the lower bound in Equation \ref{eqn:prob1:radiuscond} being
less than the upper bound in Equation \ref{eqn:prob1:bgmlv2} yields: less than the upper bound in Equation \ref{eqn:prob1:bgmlv2} yields:
\begin{equation} \begin{equation}
r<\sage{problem1.positive_intersection_bgmlv2} r<\sage{problem1.positive_intersection_bgmlv2}
\end{equation} \end{equation}
\noindent \noindent
Similarly, but with the upper bound in Equation \ref{eqn:prob1:bgmlv3}, gives: Similarly, but with the upper bound in Equation \ref{eqn:prob1:bgmlv3}, gives:
\begin{equation} \begin{equation}
r<\sage{problem1.positive_intersection_bgmlv3} r<\sage{problem1.positive_intersection_bgmlv3}
\end{equation} \end{equation}
\noindent \noindent
...@@ -703,7 +704,7 @@ bound, over $q$ in this range, to get a simpler (but weaker) bound in the ...@@ -703,7 +704,7 @@ bound, over $q$ in this range, to get a simpler (but weaker) bound in the
following Lemma \ref{lem:prob1:convenient_r_bound}. following Lemma \ref{lem:prob1:convenient_r_bound}.
\begin{theorem}[Problem \ref{problem:problem-statement-1} global upper Bound on $r$] \begin{theorem}[Problem \ref{problem:problem-statement-1} global upper Bound on $r$]
\label{lem:prob1:convenient_r_bound} \label{lem:prob1:convenient_r_bound}
Let $u$ be a solution to Problem \ref{problem:problem-statement-1}. Let $u$ be a solution to Problem \ref{problem:problem-statement-1}.
Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression: Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression:
\begin{equation} \begin{equation}
...@@ -731,7 +732,7 @@ following Lemma \ref{lem:prob1:convenient_r_bound}. ...@@ -731,7 +732,7 @@ following Lemma \ref{lem:prob1:convenient_r_bound}.
\subsection{Bounds on Semistabiliser Rank \texorpdfstring{$r$}{} in Problem \subsection{Bounds on Semistabiliser Rank \texorpdfstring{$r$}{} in Problem
\ref{problem:problem-statement-2}} \ref{problem:problem-statement-2}}
\label{subsec:bounds-on-semistab-rank-prob-2} \label{subsec:bounds-on-semistab-rank-prob-2}
Now, the inequalities from the above Subsubsection Now, the inequalities from the above Subsubsection
...@@ -763,13 +764,13 @@ $\chern^{\beta_0}(u) > 0$ ...@@ -763,13 +764,13 @@ $\chern^{\beta_0}(u) > 0$
(Equation \ref{eqn:radiuscond_d_bound_betamin}) we get: (Equation \ref{eqn:radiuscond_d_bound_betamin}) we get:
\begin{sagesilent} \begin{sagesilent}
from plots_and_expressions import \ from plots_and_expressions import \
positive_radius_condition_with_q, \ positive_radius_condition_with_q, \
q_value_expr, \ q_value_expr, \
beta_value_expr beta_value_expr
\end{sagesilent} \end{sagesilent}
\begin{equation} \begin{equation}
\label{eqn:positive_rad_condition_in_terms_of_q_beta} \label{eqn:positive_rad_condition_in_terms_of_q_beta}
\frac{1}{\lcm(m,2)}\ZZ \frac{1}{\lcm(m,2)}\ZZ
\ni \ni
\:\: \:\:
...@@ -786,10 +787,10 @@ proof of Theorem ...@@ -786,10 +787,10 @@ proof of Theorem
\begin{sagesilent} \begin{sagesilent}
from plots_and_expressions import main_theorem1, betamin_subs from plots_and_expressions import main_theorem1, betamin_subs
\end{sagesilent} \end{sagesilent}
\begin{theorem}[First bound on $r$ for Problem \ref{problem:problem-statement-2}] \begin{theorem}[First bound on $r$ for Problem \ref{problem:problem-statement-2}]
\label{thm:rmax_with_uniform_eps} \label{thm:rmax_with_uniform_eps}
Let $X$ be a smooth projective surface with Picard rank 1 and choice of ample 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 line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
$m\coloneqq\ell^2$. $m\coloneqq\ell^2$.
...@@ -803,8 +804,8 @@ from plots_and_expressions import main_theorem1, betamin_subs ...@@ -803,8 +804,8 @@ from plots_and_expressions import main_theorem1, betamin_subs
\begin{align*} \begin{align*}
\min \min
\left( \left(
\sage{main_theorem1.r_upper_bound1.subs(betamin_subs)}, \:\: \sage{main_theorem1.r_upper_bound1.subs(betamin_subs)}, \:\:
\sage{main_theorem1.r_upper_bound2.subs(betamin_subs)} \sage{main_theorem1.r_upper_bound2.subs(betamin_subs)}
\right) \right)
\end{align*} \end{align*}
\noindent \noindent
...@@ -812,57 +813,57 @@ from plots_and_expressions import main_theorem1, betamin_subs ...@@ -812,57 +813,57 @@ from plots_and_expressions import main_theorem1, betamin_subs
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}
Both $d$ and the lower bound in Both $d$ and the lower bound in
(Equation \ref{eqn:positive_rad_condition_in_terms_of_q_beta}) (Equation \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$. are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
So, if any of the two upper bounds on $d$ come to within So, if any of the two upper bounds on $d$ come to within
$\epsilon_v \coloneqq \frac{1}{\lcm(m,2n^2)}$ of this lower bound, $\epsilon_v \coloneqq \frac{1}{\lcm(m,2n^2)}$ of this lower bound,
then there are no solutions for $d$. then there are no solutions for $d$.
Hence any corresponding $r$ cannot be a rank of a Hence any corresponding $r$ cannot be a rank of a
pseudo-semistabiliser for $v$. pseudo-semistabiliser for $v$.
To avoid this, we must have, To avoid this, we must have,
considering Equations considering Equations
\ref{eqn:bgmlv2_d_bound_betamin}, \ref{eqn:bgmlv2_d_bound_betamin},
\ref{eqn:bgmlv3_d_bound_betamin}, \ref{eqn:bgmlv3_d_bound_betamin},
\ref{eqn:radiuscond_d_bound_betamin}. \ref{eqn:radiuscond_d_bound_betamin}.
\begin{sagesilent}
from plots_and_expressions import \
assymptote_gap_condition1, assymptote_gap_condition2, k
\end{sagesilent}
\begin{align}
\epsilon_v = & \sage{assymptote_gap_condition1.subs(k==1)} \\
\epsilon_v = & \sage{assymptote_gap_condition2.subs(k==1)}
\end{align}
\begin{sagesilent} \noindent
from plots_and_expressions import \ This is equivalent to:
assymptote_gap_condition1, assymptote_gap_condition2, k
\end{sagesilent}
\begin{align}
\epsilon_v =&\sage{assymptote_gap_condition1.subs(k==1)} \\
\epsilon_v =&\sage{assymptote_gap_condition2.subs(k==1)}
\end{align}
\noindent
This is equivalent to:
\begin{equation} \begin{equation}
\label{eqn:thm-bound-for-r-impossible-cond-for-r} \label{eqn:thm-bound-for-r-impossible-cond-for-r}
r \leq r \leq
\min\left( \min\left(
\sage{ \sage{
main_theorem1.r_upper_bound1 main_theorem1.r_upper_bound1
} , } ,
\sage{ \sage{
main_theorem1.r_upper_bound2 main_theorem1.r_upper_bound2
} }
\right) \right)
\end{equation} \end{equation}
\end{proof} \end{proof}
\begin{sagesilent} \begin{sagesilent}
from plots_and_expressions import q_sol, bgmlv_v, psi from plots_and_expressions import q_sol, bgmlv_v, psi
\end{sagesilent} \end{sagesilent}
\begin{corollary}[Second, global bound on $r$ for Problem \ref{problem:problem-statement-2}] \begin{corollary}[Second, global bound on $r$ for Problem \ref{problem:problem-statement-2}]
\label{cor:direct_rmax_with_uniform_eps} \label{cor:direct_rmax_with_uniform_eps}
Let $X$ be a smooth projective surface with Picard rank 1 and choice of ample 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 line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
$m\coloneqq\ell^2$. $m\coloneqq\ell^2$.
...@@ -873,90 +874,90 @@ from plots_and_expressions import q_sol, bgmlv_v, psi ...@@ -873,90 +874,90 @@ from plots_and_expressions import q_sol, bgmlv_v, psi
are bounded above as follows. are bounded above as follows.
\begin{align*} \begin{align*}
r &\leq \sage{main_theorem1.corollary_r_bound} r & \leq \sage{main_theorem1.corollary_r_bound}
&\text{if } R < \frac{\Delta(v)\lcm(m,2n^2)}{2m} & \text{if } R < \frac{\Delta(v)\lcm(m,2n^2)}{2m}
\\ \\
r &\leq \frac{\Delta(v)\lcm(m,2n^2)}{2m} r & \leq \frac{\Delta(v)\lcm(m,2n^2)}{2m}
&\text{if } R \geq \frac{\Delta(v)\lcm(m,2n^2)}{2m} & \text{if } R \geq \frac{\Delta(v)\lcm(m,2n^2)}{2m}
\end{align*} \end{align*}
\end{corollary} \end{corollary}
\begin{proof} \begin{proof}
The ranks of the pseudo-semistabilisers for $v$ are bounded above by the The ranks of the pseudo-semistabilisers for $v$ are bounded above by the
maximum over $q\in [0, \chern_1^{\beta_{-}}(v)]$ of the expression in Theorem maximum over $q\in [0, \chern_1^{\beta_{-}}(v)]$ of the expression in Theorem
\ref{thm:rmax_with_uniform_eps}. \ref{thm:rmax_with_uniform_eps}.
Noticing that the expression is a maximum of two quadratic functions in $q$ Noticing that the expression is a maximum of two quadratic functions in $q$
($\beta_0=\beta_{-}(v)$ in this context): ($\beta_0=\beta_{-}(v)$ in this context):
\begin{equation*} \begin{equation*}
f_1(q)\coloneqq\sage{main_theorem1.r_upper_bound1} \qquad f_1(q)\coloneqq\sage{main_theorem1.r_upper_bound1} \qquad
f_2(q)\coloneqq\sage{main_theorem1.r_upper_bound2} f_2(q)\coloneqq\sage{main_theorem1.r_upper_bound2}
\end{equation*} \end{equation*}
These have their minimums at $q=0$ and $q=\chern_1^{\beta_{-}}(v)$ respectively, These have their minimums at $q=0$ and $q=\chern_1^{\beta_{-}}(v)$ respectively,
with values 0 and $R>0$ respectively. with values 0 and $R>0$ respectively.
So provided that So provided that
$f_2\left(\chern^{\beta_{-}}_1(v)\right) < f_1\left(\chern^{\beta_{-}}_1(v)\right)$, $f_2\left(\chern^{\beta_{-}}_1(v)\right) < f_1\left(\chern^{\beta_{-}}_1(v)\right)$,
the maximum is achieved at their intersection. the maximum is achieved at their intersection.
Otherwise, the maximum is achieved at Otherwise, the maximum is achieved at
$\chern^{\beta_{-}}_1(v)$. $\chern^{\beta_{-}}_1(v)$.
So we can say that So we can say that
\begin{align*} \begin{align*}
r &\leq r & \leq
f_{1}(q_{\mathrm{max}}) f_{1}(q_{\mathrm{max}})
&\text{if } f_2\left(\chern^{\beta_{-}}_1(v)\right) < & \text{if } f_2\left(\chern^{\beta_{-}}_1(v)\right) <
f_1\left(\chern^{\beta_{-}}_1(v)\right) f_1\left(\chern^{\beta_{-}}_1(v)\right)
\\ && \\ &&
\text{where $q_{\mathrm{max}}$ is the $q$-value where the $f_i$ intersect} \text{where $q_{\mathrm{max}}$ is the $q$-value where the $f_i$ intersect}
\\ \\
r &\leq f_1\left(\chern^{\beta_{-}}(v)\right) r & \leq f_1\left(\chern^{\beta_{-}}(v)\right)
&\text{if } f_2\left(\chern^{\beta_{-}}_1(v)\right) \geq & \text{if } f_2\left(\chern^{\beta_{-}}_1(v)\right) \geq
f_1\left(\chern^{\beta_{-}}_1(v)\right) f_1\left(\chern^{\beta_{-}}_1(v)\right)
\end{align*} \end{align*}
\noindent \noindent
In the first case, In the first case,
solving for $f_1(q)=f_2(q)$ yields solving for $f_1(q)=f_2(q)$ yields
\begin{equation*} \begin{equation*}
q=\sage{q_sol.expand()} q=\sage{q_sol.expand()}
\end{equation*} \end{equation*}
And evaluating $f_1$ at this $q$-value gives: And evaluating $f_1$ at this $q$-value gives:
\begin{equation*} \begin{equation*}
\sage{main_theorem1.corollary_intermediate} \sage{main_theorem1.corollary_intermediate}
\end{equation*} \end{equation*}
\noindent \noindent
Finally, noting that $\Delta(v)=\left(\chern_1^{\beta_{-}(v)}(v)\right)^2\ell^2$, Finally, noting that $\Delta(v)=\left(\chern_1^{\beta_{-}(v)}(v)\right)^2\ell^2$,
we get the bounds as stated in the statement of the Corollary. we get the bounds as stated in the statement of the Corollary.
\end{proof} \end{proof}
\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$] \begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-second} \label{exmpl:recurring-second}
Just like in Example \ref{exmpl:recurring-first}, take Just like in Example \ref{exmpl:recurring-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{recurring.betaminus}$, that $m=2$, $\beta=\sage{recurring.betaminus}$,
giving $n=\sage{recurring.n}$. giving $n=\sage{recurring.n}$.
Using the above Corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that Using the above Corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilisers for $v$ are bounded above by the ranks of tilt semistabilisers for $v$ are bounded above by
$\sage{recurring.corrolary_bound} \approx $\sage{recurring.corrolary_bound} \approx
\sage{round(float(recurring.corrolary_bound), 1)}$, \sage{round(float(recurring.corrolary_bound), 1)}$,
which is much closer to real maximum 25 than the original bound 144. which is much closer to real maximum 25 than the original bound 144.
\end{example} \end{example}
\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$] \begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-second} \label{exmpl:extravagant-second}
Just like in Example \ref{exmpl:extravagant-first}, take Just like in Example \ref{exmpl:extravagant-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{extravagant.betaminus}$, that $m=2$, $\beta=\sage{extravagant.betaminus}$,
giving $n=\sage{extravagant.n}$. giving $n=\sage{extravagant.n}$.
Using the above Corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that Using the above Corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilisers for $v$ are bounded above by the ranks of tilt semistabilisers for $v$ are bounded above by
$\sage{extravagant.corrolary_bound} \approx $\sage{extravagant.corrolary_bound} \approx
\sage{round(float(extravagant.corrolary_bound), 1)}$, \sage{round(float(extravagant.corrolary_bound), 1)}$,
which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
original bound 215296. original bound 215296.
\end{example} \end{example}
%% refinements using specific values of q and beta %% refinements using specific values of q and beta
...@@ -982,8 +983,8 @@ Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_v$ in the ...@@ -982,8 +983,8 @@ Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_v$ in the
proof of Theorem \ref{thm:rmax_with_uniform_eps}: proof of Theorem \ref{thm:rmax_with_uniform_eps}:
\begin{lemmadfn}[% \begin{lemmadfn}[%
A better alternative to $\epsilon_v$: A better alternative to $\epsilon_v$:
$\epsilon_{v,q}$ $\epsilon_{v,q}$
] ]
\label{lemdfn:epsilon_q} \label{lemdfn:epsilon_q}
Suppose $d \in \frac{1}{\lcm(m,2n^2)}\ZZ$ satisfies the condition in Suppose $d \in \frac{1}{\lcm(m,2n^2)}\ZZ$ satisfies the condition in
...@@ -1021,94 +1022,94 @@ proof of Theorem \ref{thm:rmax_with_uniform_eps}: ...@@ -1021,94 +1022,94 @@ proof of Theorem \ref{thm:rmax_with_uniform_eps}:
\mod{\gcd\left( \mod{\gcd\left(
\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)}, \frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
\frac{mn\aa}{\gcd(m,2n^2)} \frac{mn\aa}{\gcd(m,2n^2)}
\right)} \right)}
\end{equation*} \end{equation*}
\end{lemmadfn} \end{lemmadfn}
\begin{remark} \begin{remark}
The quantity $m$ is determined by the variety, whereas $a_v$ and $n$ are determined by the Chern 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. 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 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 constant in the context of the problem we are solving for
(i.e. Problem \ref{problem:problem-statement-2}). (i.e. Problem \ref{problem:problem-statement-2}).
However $b_q$ depends on the choice of $q$, that is the value of 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 $\chern_1^{\beta_{-}(v)}(u)$ 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. why $k_{v,q}$ is denoted to depend on $q$ on top of $v$ and the context of the problem.
\end{remark} \end{remark}
\begin{proof} \begin{proof}
Consider the following sequence of logical implications. Consider the following sequence of logical implications.
The one-way implication follows from The one-way implication follows from
$\aa r + \bb \equiv 0 \pmod{n}$, $\aa r + \bb \equiv 0 \pmod{n}$,
and the final logical equivalence is just a simplification of the expressions. and the final logical equivalence is just a simplification of the expressions.
\begin{align} \begin{align}
\frac{ x }{ \lcm(m,2) } \frac{ x }{ \lcm(m,2) }
- \frac{ - \frac{
(\aa r+2\bb)\aa (\aa r+2\bb)\aa
}{ }{
2n^2 2n^2
} }
= \frac{ k }{ \lcm(m,2n^2) } = \frac{ k }{ \lcm(m,2n^2) }
\quad \text{for some } x \in \ZZ \quad \text{for some } x \in \ZZ
\span \span \span \span \span \span \span \span \span \span
\label{eqn:finding_better_eps_problem} \label{eqn:finding_better_eps_problem}
\\ \nonumber \\ \nonumber
\\ \Leftrightarrow& & \\ \Leftrightarrow& &
- (\aa r+2\bb)\aa - (\aa r+2\bb)\aa
\frac{\lcm(m,2n^2)}{2n^2} \frac{\lcm(m,2n^2)}{2n^2}
&\equiv k && & \equiv k & &
\nonumber \nonumber
\\ &&& \\ &&&
\mod \frac{\lcm(m,2n^2)}{\lcm(m,2)} \mod \frac{\lcm(m,2n^2)}{\lcm(m,2)}
\span \span \span \span \span \span
\nonumber \nonumber
\\ \Rightarrow& & \\ \Rightarrow& &
- \bb\aa - \bb\aa
\frac{\lcm(m,2n^2)}{2n^2} \frac{\lcm(m,2n^2)}{2n^2}
&\equiv k && & \equiv k & &
\nonumber \nonumber
\\ &&& \\ &&&
\mod \gcd\left( \mod \gcd\left(
\frac{\lcm(m,2n^2)}{\lcm(m,2)}, \frac{\lcm(m,2n^2)}{\lcm(m,2)},
\frac{n \aa \lcm(m,2n^2)}{2n^2} \frac{n \aa \lcm(m,2n^2)}{2n^2}
\right) \right)
\span \span \span \span \span \span
\nonumber \nonumber
\\ \Leftrightarrow& & \\ \Leftrightarrow& &
- \bb\aa - \bb\aa
\frac{m}{\gcd(m,2n^2)} \frac{m}{\gcd(m,2n^2)}
&\equiv k && & \equiv k & &
\label{eqn:better_eps_problem_k_mod_n} \label{eqn:better_eps_problem_k_mod_n}
\\ &&& \\ &&&
\mod \gcd\left( \mod \gcd\left(
\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)}, \frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
\frac{mn \aa}{\gcd(m,2n^2)} \frac{mn \aa}{\gcd(m,2n^2)}
\right) \right)
\span \span \span \span \span \span
\nonumber \nonumber
\end{align} \end{align}
In our situation, we want to find the least $k>0$ satisfying In our situation, we want to find the least $k>0$ satisfying
Equation \ref{eqn:finding_better_eps_problem}. Equation \ref{eqn:finding_better_eps_problem}.
Since such a $k$ must also satisfy Equation \ref{eqn:better_eps_problem_k_mod_n}, 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 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 $r$). (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 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:finding_better_eps_problem}, giving the first inequality in Equation
\ref{eqn:epsilon_q_lemma_prop}. \ref{eqn:epsilon_q_lemma_prop}.
Furthermore, $k_{v,q}\geq 1$ gives the second part of the inequality: Furthermore, $k_{v,q}\geq 1$ gives the second part of the inequality:
$\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$. $\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.
\end{proof} \end{proof}
\begin{sagesilent} \begin{sagesilent}
from plots_and_expressions import main_theorem2 from plots_and_expressions import main_theorem2
\end{sagesilent} \end{sagesilent}
\begin{theorem}[Third bound on $r$ for Problem \ref{problem:problem-statement-2}] \begin{theorem}[Third bound on $r$ for Problem \ref{problem:problem-statement-2}]
\label{thm:rmax_with_eps1} \label{thm:rmax_with_eps1}
Let $X$ be a smooth projective surface with Picard rank 1 and choice of ample 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 line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
$m\coloneqq\ell^2$. $m\coloneqq\ell^2$.
...@@ -1122,8 +1123,8 @@ from plots_and_expressions import main_theorem2 ...@@ -1122,8 +1123,8 @@ from plots_and_expressions import main_theorem2
\begin{align*} \begin{align*}
\min \min
\left( \left(
\sage{main_theorem2.r_upper_bound1.subs(betamin_subs)}, \:\: \sage{main_theorem2.r_upper_bound1.subs(betamin_subs)}, \:\:
\sage{main_theorem2.r_upper_bound2.subs(betamin_subs)} \sage{main_theorem2.r_upper_bound2.subs(betamin_subs)}
\right), \right),
\end{align*} \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},
...@@ -1142,10 +1143,10 @@ Although the general form of this bound is quite complicated, it does simplify a ...@@ -1142,10 +1143,10 @@ Although the general form of this bound is quite complicated, it does simplify a
lot when $m$ is small. lot when $m$ is small.
\begin{sagesilent} \begin{sagesilent}
from plots_and_expressions import main_theorem2_corollary from plots_and_expressions import main_theorem2_corollary
\end{sagesilent} \end{sagesilent}
\begin{corollary}[Third bound on $r$ on $\PP^2$ and principally polarised abelian surfaces] \begin{corollary}[Third bound on $r$ on $\PP^2$ and principally polarised abelian surfaces]
\label{cor:rmax_with_eps1} \label{cor:rmax_with_eps1}
Suppose we are working over $\PP^2$ or a principally polarised abelian surface Suppose we are working over $\PP^2$ or a principally polarised abelian surface
(or any other surfaces with $m=\ell^2=1$ or $2$). (or any other surfaces with $m=\ell^2=1$ or $2$).
Let $v$ be a fixed Chern character, with $\beta_{-}\coloneqq\beta_{-}(v)=\frac{a_v}{n}$ Let $v$ be a fixed Chern character, with $\beta_{-}\coloneqq\beta_{-}(v)=\frac{a_v}{n}$
...@@ -1158,127 +1159,127 @@ from plots_and_expressions import main_theorem2_corollary ...@@ -1158,127 +1159,127 @@ from plots_and_expressions import main_theorem2_corollary
\begin{align*} \begin{align*}
\min \min
\left( \left(
\sage{main_theorem2_corollary.r_upper_bound1.subs(betamin_subs)}, \:\: \sage{main_theorem2_corollary.r_upper_bound1.subs(betamin_subs)}, \:\:
\sage{main_theorem2_corollary.r_upper_bound2.subs(betamin_subs)} \sage{main_theorem2_corollary.r_upper_bound2.subs(betamin_subs)}
\right), \right),
\end{align*} \end{align*}
where $R = \chern_0(v)$ and $k_{v,q}$ is the least where $R = \chern_0(v)$ and $k_{v,q}$ is the least
$k\in\ZZ_{>0}$ satisfying $k\in\ZZ_{>0}$ satisfying
${ ${
k \equiv -\aa\bb k \equiv -\aa\bb
\pmod{n} \pmod{n}
}$. }$.
\end{corollary} \end{corollary}
\begin{proof} \begin{proof}
This is a specialisation of Theorem \ref{thm:rmax_with_eps1}, where we can 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 drastically simplify the $\lcm$ and $\gcd$ terms by noting that $m$ divides both
$2$ and $2n^2$, and that $a_v$ is coprime to $n$. $2$ and $2n^2$, and that $a_v$ is coprime to $n$.
\end{proof} \end{proof}
\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$] \begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-third} \label{exmpl:recurring-third}
Just like in Examples \ref{exmpl:recurring-first} and Just like in Examples \ref{exmpl:recurring-first} and
\ref{exmpl:recurring-second}, \ref{exmpl:recurring-second},
take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
$\beta=\sage{recurring.betaminus}$, giving $n=\sage{recurring.n}$ $\beta=\sage{recurring.betaminus}$, giving $n=\sage{recurring.n}$
and $\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$. and $\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
%% TODO transcode notebook code %% TODO transcode notebook code
The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabiliser $u$ of $v$ 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: in terms of the possible values for $q\coloneqq\chern_1^{\beta_{-}}(u)$ are as follows:
\begin{sagesilent} \begin{sagesilent}
from examples import bound_comparisons from examples import bound_comparisons
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring) qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
\end{sagesilent} \end{sagesilent}
\vspace{1em} \vspace{1em}
\noindent \noindent
\directlua{ table_width = 3*4+1 } \directlua{ table_width = 3*4+1 }
\begin{tabular}{l\directlua{for i=0,table_width-1 do tex.sprint([[|c]]) end}} \begin{tabular}{l\directlua{for i=0,table_width-1 do tex.sprint([[|c]]) end}}
$q=\chern_1^{\beta_{-}}(u)$ $q=\chern_1^{\beta_{-}}(u)$
\directlua{for i=0,table_width-1 do \directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$" local cell = [[ & $\noexpand\sage{qs[]] .. i .. "]}$"
tex.sprint(cell) tex.sprint(cell)
end} end}
\\ \hline \\ \hline
Theorem \ref{thm:rmax_with_uniform_eps} Theorem \ref{thm:rmax_with_uniform_eps}
\directlua{for i=0,table_width-1 do \directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$" local cell = [[ & $\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
tex.sprint(cell) tex.sprint(cell)
end} end}
\\ \\
Theorem \ref{thm:rmax_with_eps1} Theorem \ref{thm:rmax_with_eps1}
\directlua{for i=0,table_width-1 do \directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$" local cell = [[ & $\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
tex.sprint(cell) tex.sprint(cell)
end} end}
\end{tabular} \end{tabular}
\vspace{1em} \vspace{1em}
\noindent \noindent
It is worth noting that the bounds given by Theorem \ref{thm:rmax_with_eps1} 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 reach, but do not exceed, the actual maximum rank 25 of the
pseudo-semistabilisers of $v$ in this case. pseudo-semistabilisers of $v$ in this case.
As a reminder, the original loose bound from Theorem \ref{thm:loose-bound-on-r} As a reminder, the original loose bound from Theorem \ref{thm:loose-bound-on-r}
was 144. was 144.
\end{example} \end{example}
\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$] \begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-third} \label{exmpl:extravagant-third}
Just like in examples \ref{exmpl:extravagant-first} and Just like in examples \ref{exmpl:extravagant-first} and
\ref{exmpl:extravagant-second}, \ref{exmpl:extravagant-second},
take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
$\beta=\sage{extravagant.betaminus}$, giving $n=\sage{n:=extravagant.n}$ $\beta=\sage{extravagant.betaminus}$, giving $n=\sage{n:=extravagant.n}$
and $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$. 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 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 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 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 \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. 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 $v$ 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: in terms of the first few smallest possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
\begin{sagesilent} \begin{sagesilent}
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(extravagant) qs, theorem2_bounds, theorem3_bounds = bound_comparisons(extravagant)
\end{sagesilent} \end{sagesilent}
\vspace{1em} \vspace{1em}
\noindent \noindent
\directlua{ table_width = 12 } \directlua{ table_width = 12 }
\begin{tabular}{l\directlua{for i=0,table_width do tex.sprint([[|c]]) end}} \begin{tabular}{l\directlua{for i=0,table_width do tex.sprint([[|c]]) end}}
$q=\chern_1^\beta(u)$ $q=\chern_1^\beta(u)$
\directlua{for i=0,table_width-1 do \directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$" local cell = [[ & $\noexpand\sage{qs[]] .. i .. "]}$"
tex.sprint(cell) tex.sprint(cell)
end} end}
&$\cdots$ & $\cdots$
\\ \hline \\ \hline
Theorem \ref{thm:rmax_with_uniform_eps} Theorem \ref{thm:rmax_with_uniform_eps}
\directlua{for i=0,table_width-1 do \directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$" local cell = [[ & $\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
tex.sprint(cell) tex.sprint(cell)
end} end}
&$\cdots$ & $\cdots$
\\ \\
Theorem \ref{thm:rmax_with_eps1} Theorem \ref{thm:rmax_with_eps1}
\directlua{for i=0,table_width-1 do \directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$" local cell = [[ & $\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
tex.sprint(cell) tex.sprint(cell)
end} end}
&$\cdots$ & $\cdots$
\end{tabular} \end{tabular}
\vspace{1em} \vspace{1em}
\noindent \noindent
However the reduction in the overall bound on $r$ is not as drastic, since all 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 possible values for $k_{v,q}$ in $\{1,2,\ldots,\sage{n}\}$ are iterated through
cyclically as we consider successive possible values for $q$. cyclically as we consider successive possible values for $q$.
And for each $q$ where $k_{v,q}=1$, both theorems give the same bound. And for each $q$ where $k_{v,q}=1$, both theorems give the same bound.
Calculating the maximums over all values of $q$ yields Calculating the maximums over all values of $q$ yields
$\sage{max(theorem2_bounds)}$ for Theorem \ref{thm:rmax_with_uniform_eps}, and $\sage{max(theorem2_bounds)}$ for Theorem \ref{thm:rmax_with_uniform_eps}, and
$\sage{max(theorem3_bounds)}$ for Theorem \ref{thm:rmax_with_eps1}. $\sage{max(theorem3_bounds)}$ for Theorem \ref{thm:rmax_with_eps1}.
\end{example} \end{example}
tex/computing-solutions.tex
View file @ 1a27cd54
...@@ -13,7 +13,7 @@ a different algorithm will be presented making use of theorems from Section ...@@ -13,7 +13,7 @@ a different algorithm will be presented making use of theorems from Section
with the goal of cutting down the run time. with the goal of cutting down the run time.
\subsubsection{Finding possible \texorpdfstring{$r$}{r} and \subsubsection{Finding possible \texorpdfstring{$r$}{r} and
\texorpdfstring{$c$}{c}} \texorpdfstring{$c$}{c}}
To do this, first calculate the upper bound $r_{\mathrm{max}}$ on the ranks of tilt To do this, first calculate the upper bound $r_{\mathrm{max}}$ on the ranks of tilt
semistabilisers, as given by Theorem \ref{thm:loose-bound-on-r}. semistabilisers, as given by Theorem \ref{thm:loose-bound-on-r}.
...@@ -36,7 +36,7 @@ the Bogomolov inequalities and Consequence 3 of Lemma ...@@ -36,7 +36,7 @@ the Bogomolov inequalities and Consequence 3 of Lemma
($\chern_2^{\beta_{-}}(u)>0$). ($\chern_2^{\beta_{-}}(u)>0$).
\subsubsection{Finding \texorpdfstring{$d$}{d} for fixed \texorpdfstring{$r$}{r} \subsubsection{Finding \texorpdfstring{$d$}{d} for fixed \texorpdfstring{$r$}{r}
and \texorpdfstring{$c$}{c}} and \texorpdfstring{$c$}{c}}
$\Delta(u) \geq 0$ induces an upper bound $\frac{c^2}{2r}$ on $d$, and the $\Delta(u) \geq 0$ induces an upper bound $\frac{c^2}{2r}$ on $d$, and the
$\chern_2^{\beta_{-}}(u)>0$ condition induces a lower bound on $d$. $\chern_2^{\beta_{-}}(u)>0$ condition induces a lower bound on $d$.
...@@ -51,8 +51,8 @@ end up not yielding any solutions for the problem. ...@@ -51,8 +51,8 @@ end up not yielding any solutions for the problem.
In fact, solutions $u$ to our problem with high rank must have $\mu(u)$ close to In fact, solutions $u$ to our problem with high rank must have $\mu(u)$ close to
$\beta_{-}(v)$: $\beta_{-}(v)$:
\begin{align*} \begin{align*}
0 &\leq \chern_1^{\beta_{-}}(u) \leq \chern_1^{\beta_{-}}(u) \\ 0 & \leq \chern_1^{\beta_{-}}(u) \leq \chern_1^{\beta_{-}}(u) \\
0 &\leq \mu(u) - \beta_{-} \leq \frac{\chern_1^{\beta_{-}}(v)}{r} 0 & \leq \mu(u) - \beta_{-} \leq \frac{\chern_1^{\beta_{-}}(v)}{r}
\end{align*} \end{align*}
In particular, it is the $\chern_1^{\beta_{-}}(v-u) \geq 0$ condition which In particular, it is the $\chern_1^{\beta_{-}}(v-u) \geq 0$ condition which
fails for $r$,$c$ pairs with large $r$ and $\frac{c}{r}$ too far from $\beta_{-}$. fails for $r$,$c$ pairs with large $r$ and $\frac{c}{r}$ too far from $\beta_{-}$.
...@@ -66,17 +66,17 @@ alternative algorithm which will later be described in Section ...@@ -66,17 +66,17 @@ alternative algorithm which will later be described in Section
\ref{sect:prob2-algorithm}. \ref{sect:prob2-algorithm}.
\begin{center} \begin{center}
\label{table:bench-schmidt-vs-nay} \label{table:bench-schmidt-vs-nay}
\begin{tabular}{ |r|l|l| } \begin{tabular}{ |r|l|l| }
\hline \hline
Choice of $v$ on $\mathbb{P}^2$ Choice of $v$ on $\mathbb{P}^2$
& $(3, 2\ell, -2)$ & $(3, 2\ell, -2)$
& $(3, 2\ell, -\frac{15}{2})$ \\ & $(3, 2\ell, -\frac{15}{2})$ \\
\hline \hline
\cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s & >1hr \\ \cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s & >1hr \\
\cite{NaylorRust2023} exec time & \sim 50ms & \sim 50ms \\ \cite{NaylorRust2023} exec time & \sim 50ms & \sim 50ms \\
\hline \hline
\end{tabular} \end{tabular}
\end{center} \end{center}
\section{Computing Solutions to Problem \ref{problem:problem-statement-2}} \section{Computing Solutions to Problem \ref{problem:problem-statement-2}}
...@@ -94,7 +94,7 @@ The algorithm yields solutions ...@@ -94,7 +94,7 @@ The algorithm yields solutions
$u=(r,c\ell,d\ell^2)$ to the problem as follows. $u=(r,c\ell,d\ell^2)$ to the problem as follows.
\subsubsection{Iterating Over Possible \subsubsection{Iterating Over Possible
\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}} \texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}}
Given a Chern character $v$, the domain of the problem are first verified: that Given a Chern character $v$, the domain of the problem are first verified: that
$v$ has positive rank, that it satisfies $\Delta(v) \geq 0$, and that $v$ has positive rank, that it satisfies $\Delta(v) \geq 0$, and that
...@@ -102,6 +102,7 @@ $\beta_{-}(v)$ is rational. ...@@ -102,6 +102,7 @@ $\beta_{-}(v)$ is rational.
Take $\beta_{-}(v)=\frac{a_v}{n}$ in simplest terms. Take $\beta_{-}(v)=\frac{a_v}{n}$ in simplest terms.
Iterate over $q = \frac{b_q}{n} \in (0,\chern_1^{\beta_{-}}(v))\cap\frac{1}{n}\ZZ$. Iterate over $q = \frac{b_q}{n} \in (0,\chern_1^{\beta_{-}}(v))\cap\frac{1}{n}\ZZ$.
The code used to generate the corresponding values for $b_q$ is shown in Listing The code used to generate the corresponding values for $b_q$ is shown in Listing
% texlab: ignore
\ref{fig:code:consideredb}. \ref{fig:code:consideredb}.
\lstinputlisting[ \lstinputlisting[
...@@ -118,6 +119,7 @@ We can therefore reduce the problem of finding solutions to the more specialised ...@@ -118,6 +119,7 @@ We can therefore reduce the problem of finding solutions to the more specialised
problem of finding the solutions $u$ with each fixed possible $q=\chern_1^\beta(u)$ problem of finding the solutions $u$ with each fixed possible $q=\chern_1^\beta(u)$
(i.e. choice of $b$). (i.e. choice of $b$).
The code representing this appears in Listing The code representing this appears in Listing
% texlab: ignore
\ref{fig:code:reducingtoeachb}. \ref{fig:code:reducingtoeachb}.
Line 16 refers to creating an objects representing the context the specialised Line 16 refers to creating an objects representing the context the specialised
problem for the fixed $q$ value, with the next line `solving' the specialised problem for the fixed $q$ value, with the next line `solving' the specialised
...@@ -133,9 +135,9 @@ and collect up the results. ...@@ -133,9 +135,9 @@ and collect up the results.
]{../tilt.rs/src/tilt_stability/find_all.git-untrack.rs.tex.git-untrack} ]{../tilt.rs/src/tilt_stability/find_all.git-untrack.rs.tex.git-untrack}
\subsubsection{Iterating Over Possible \subsubsection{Iterating Over Possible
\texorpdfstring{$r=\chern_0(u)$}{r} \texorpdfstring{$r=\chern_0(u)$}{r}
for Fixed for Fixed
\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q} \texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
} }
Let $q=\frac{b_q}{n}$, for which we are now solving the more specialised problem of finding Let $q=\frac{b_q}{n}$, for which we are now solving the more specialised problem of finding
...@@ -159,6 +161,7 @@ Fixing $r$ and $q$ also determines $c\coloneqq\chern_1(u)$, and so we can genera ...@@ -159,6 +161,7 @@ Fixing $r$ and $q$ also determines $c\coloneqq\chern_1(u)$, and so we can genera
the corresponding values of $c$, as we generate the $r$ values. the corresponding values of $c$, as we generate the $r$ values.
It now remains to solve the problem for each of the combinations of fixed values It now remains to solve the problem for each of the combinations of fixed values
for $q$ and $r$ (and consequently $c$) considered. for $q$ and $r$ (and consequently $c$) considered.
% texlab: ignore
This is shown in Listing \ref{fig:code:reducingtoeachr}. This is shown in Listing \ref{fig:code:reducingtoeachr}.
\lstinputlisting[ \lstinputlisting[
...@@ -170,11 +173,11 @@ This is shown in Listing \ref{fig:code:reducingtoeachr}. ...@@ -170,11 +173,11 @@ This is shown in Listing \ref{fig:code:reducingtoeachr}.
\subsubsection{Iterating Over Possible \subsubsection{Iterating Over Possible
\texorpdfstring{$d=\chern_2(u)/\ell^2$}{d} \texorpdfstring{$d=\chern_2(u)/\ell^2$}{d}
for Fixed for Fixed
\texorpdfstring{$r=\chern_0(u)$}{r} \texorpdfstring{$r=\chern_0(u)$}{r}
and and
\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q} \texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
} }
At this point we are considering a specialisation of the problem At this point we are considering a specialisation of the problem
...@@ -198,8 +201,10 @@ equivalent to bounds on $d$ given by the equations ...@@ -198,8 +201,10 @@ equivalent to bounds on $d$ given by the equations
in Subsubsection \ref{subsubsect:all-bounds-on-d-prob2} in Subsubsection \ref{subsubsect:all-bounds-on-d-prob2}
It therefore remains to just pick values It therefore remains to just pick values
$d\in\frac{1}{\lcm(m,2n^2)}\ZZ$ within the bounds. $d\in\frac{1}{\lcm(m,2n^2)}\ZZ$ within the bounds.
% texlab: ignore
Listing \ref{fig:code:solveforfixedr} is the code for solving this Listing \ref{fig:code:solveforfixedr} is the code for solving this
specialisation of the problem, where the possible $d$ values are computed in specialisation of the problem, where the possible $d$ values are computed in
% texlab: ignore
Listing \ref{fig:code:possible_chern2}. Listing \ref{fig:code:possible_chern2}.
The explicit code for the bounds can be found in Appendix The explicit code for the bounds can be found in Appendix
\ref{appendix:subsubsec:fixed-r}. \ref{appendix:subsubsec:fixed-r}.
...@@ -232,11 +237,11 @@ decrease in computational time to find the solutions to the problem. ...@@ -232,11 +237,11 @@ decrease in computational time to find the solutions to the problem.
This could be due to a range of potential reasons: This could be due to a range of potential reasons:
\begin{itemize} \begin{itemize}
\item Unexpected optimisations from the compiler for a certain form of the \item Unexpected optimisations from the compiler for a certain form of the
program. program.
\item Increased complexity to computing the formulae for the tighter bounds. \item Increased complexity to computing the formulae for the tighter bounds.
\item Modern CPU architecture such as branch predictors \item Modern CPU architecture such as branch predictors
\cite{BranchPredictor2024} may offset the overhead of considering ranks that \cite{BranchPredictor2024} may offset the overhead of considering ranks that
turn out to be too large to have any solutions. turn out to be too large to have any solutions.
\end{itemize} \end{itemize}
For relatively small Chern characters (as those appearing in examples so far), For relatively small Chern characters (as those appearing in examples so far),
......