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tex/bounds-on-semistabilisers.tex
View file @ 1a27cd54
......@@ -4,6 +4,7 @@
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{SchmidtGithub2020} as shown in the following Listing
% texlab: ignore
\ref{fig:code:schmidt-bound}.
The latter citation is a SageMath \cite{sagemath}
library for computing certain quantities related to Bridgeland stabilities on
......@@ -18,130 +19,130 @@ pseudo-semistabilisers for tilt stability.
]{schmidt-snippet}
\begin{theorem}[Bound on $r$ - Benjamin Schmidt]
\label{thm:loose-bound-on-r}
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
take $m\coloneqq \ell^2$.
Given a Chern character $v$ with $\Delta(v) \geq 0$ and positive rank (or
$\chern_0(v) = 0$ and $\chern_1(v) > 0$)
such that
$\beta_-\coloneqq\beta_{-}(v)\in\QQ$, the rank $r$ of
any solution $u$ of Problem \ref{problem:problem-statement-2} is
bounded above by:
\begin{equation*}
r \leq \frac{m\left(n \chern^{\beta_-}_1(v) - 1\right)^2}{\gcd(m,2n^2)}
\end{equation*}
\label{thm:loose-bound-on-r}
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
take $m\coloneqq \ell^2$.
Given a Chern character $v$ with $\Delta(v) \geq 0$ and positive rank (or
$\chern_0(v) = 0$ and $\chern_1(v) > 0$)
such that
$\beta_-\coloneqq\beta_{-}(v)\in\QQ$, the rank $r$ of
any solution $u$ of Problem \ref{problem:problem-statement-2} is
bounded above by:
\begin{equation*}
r \leq \frac{m\left(n \chern^{\beta_-}_1(v) - 1\right)^2}{\gcd(m,2n^2)}
\end{equation*}
\end{theorem}
\begin{proof}
The Bogomolov form applied to the twisted Chern character is the same as the
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}
The Bogomolov form applied to the twisted Chern character is the same as the
untwisted one.
\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}
\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
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
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
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:
\[
\chern_0(u)
\leq \frac{m\left(n \chern^{\beta_-}_1(v) - 1\right)^2}{\gcd(m,2n^2)}
\]
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}
\begin{sagesilent}
from examples import recurring
from examples import recurring
\end{sagesilent}
\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-first}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=1$, $\beta_-=\sage{recurring.betaminus}$,
giving $n=\sage{recurring.n}$ and
$\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
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
rank that appears turns out to be 25. This will be a recurring example to
illustrate the performance of later theorems about rank bounds
\label{exmpl:recurring-first}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=1$, $\beta_-=\sage{recurring.betaminus}$,
giving $n=\sage{recurring.n}$ and
$\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
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
rank that appears turns out to be 25. This will be a recurring example to
illustrate the performance of later theorems about rank bounds
\end{example}
\begin{sagesilent}
from examples import extravagant
from examples import extravagant
\end{sagesilent}
\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-first}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=1$, $\beta_-=\sage{extravagant.betaminus}$,
giving $n=\sage{extravagant.n}$ and
$\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
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
rank that appears turns out to be $\sage{round(extravagant.actual_rmax, 1)}$.
\label{exmpl:extravagant-first}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=1$, $\beta_-=\sage{extravagant.betaminus}$,
giving $n=\sage{extravagant.n}$ and
$\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
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
rank that appears turns out to be $\sage{round(extravagant.actual_rmax, 1)}$.
\end{example}
......@@ -166,19 +167,19 @@ and Corollary \ref{cor:num_test_prob2}, in a way which better fits our direction
of travel.
\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}),
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
If $u$ is a solution to the Problem then $u$ satisfies:
\begin{equation}
q\coloneqq \chern^{\beta_0}_1(u) \in \left( 0, \chern_1^{\beta_0}(v) \right)
\label{lem:eqn:cond-for-fixed-q}
\qquad
\text{and}
\qquad
\qquad
\text{and}
\qquad
\chern_0(u) > \frac{q}{\mu(v) - \beta_0}.
\nonumber
\end{equation}
......@@ -189,11 +190,11 @@ of travel.
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:
\begin{multicols}{3}
\begin{itemize}
\item $\Delta(u) \geq 0$
\item $\Delta(v-u) \geq 0$
\item $\chern^{\alpha_0,\beta_0}_2(u) \geq 0$
\end{itemize}
\begin{itemize}
\item $\Delta(u) \geq 0$
\item $\Delta(v-u) \geq 0$
\item $\chern^{\alpha_0,\beta_0}_2(u) \geq 0$
\end{itemize}
\end{multicols}
\end{lemma}
......@@ -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 $d\in \frac{1}{\lcm(m,2)}\ZZ$
which satisfy six numerical conditions.
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 first numerical condition.
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
$u=(r,c\ell,d\ell^2)$ 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
1, 2, 3 and 6 are satisfied.
$u=(r,c\ell,d\ell^2)$ 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
1, 2, 3 and 6 are satisfied.
This is in essence the second part of the Lemma.
\end{proof}
\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}),
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
rational, and written $\beta_0=\frac{a_v}{n}$ for
some coprime integers $a_v$, $n$ with $n>0$.
......@@ -227,27 +228,27 @@ of travel.
\begin{align*}
\chern^{\beta_0}_1(u)
= \frac{b_q}{n},
\qquad
a_v r &\equiv -b_q \pmod{n},
\qquad
a_v r & \equiv -b_q \pmod{n},
\quad
\text{and}
\qquad
r > \frac{q}{\mu(v) - \beta_0}
\end{align*}
\[
\text{for some }
b_q \in \left\{ 1, 2, \ldots, n\chern_1^{\beta_0}(v) - 1 \right\}.
\text{for some }
b_q \in \left\{ 1, 2, \ldots, n\chern_1^{\beta_0}(v) - 1 \right\}.
\]
And any $u = (r,c\ell,d\ell^2)$
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,
the following are satisfied:
\begin{multicols}{3}
\begin{itemize}
\item $\Delta(u) \geq 0$
\item $\Delta(v-u) \geq 0$
\item $\chern^P_2(u) \geq 0$
\end{itemize}
\begin{itemize}
\item $\Delta(u) \geq 0$
\item $\Delta(v-u) \geq 0$
\item $\chern^P_2(u) \geq 0$
\end{itemize}
\end{multicols}
\end{corollary}
......@@ -256,24 +257,24 @@ of travel.
This is a specialisation of Lemma \ref{lem:fixed-q-semistabs-criterion}
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.
Taking $\beta_0 = \frac{a_v}{n}$ we have:
for the case where $\beta_0$ is rational.
Taking $\beta_0 = \frac{a_v}{n}$ we have:
\[
q\coloneqq\chern_1^{\beta_0}(u)
= c - \frac{a_v}{n}r
\in \frac{1}{n}\ZZ
q\coloneqq\chern_1^{\beta_0}(u)
= c - \frac{a_v}{n}r
\in \frac{1}{n}\ZZ
\]
So $q=\frac{b_q}{n}$ for some $b_q \in \left\{ 1, 2, \ldots, n\chern_1^{\beta_0}(v) - 1 \right\}$
and then
${
nc - a_v r = b_q
}$
and so
nc - a_v r = b_q
}$
and so
${
a_v r \equiv -b_q
}$ modulo $n$.
a_v r \equiv -b_q
}$ modulo $n$.
\end{proof}
\subsection{Bounds on \texorpdfstring{`$d$'}{d}-values for Solutions of Problems}
......@@ -299,7 +300,7 @@ to the Problem satisfies
q \coloneqq \chern_1^{\beta_0}(u)
\in
\left(
0, \chern_1^{\beta_0}(v)
0, \chern_1^{\beta_0}(v)
\right)
\]
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,
when rearranged to a bound on $d$, amounts to:
\begin{equation}
\label{eqn:radius-cond-betamin}
\label{eqn:radius-cond-betamin}
\chern_2^{\beta_{-}}(u) > 0
\qquad
\text{and}
......@@ -332,7 +333,7 @@ when rearranged to a bound on $d$, amounts to:
\end{equation}
\begin{sagesilent}
import other_P_choice as problem1
import other_P_choice as problem1
\end{sagesilent}
In the case where we are tackling Problem \ref{problem:problem-statement-1},
......@@ -368,7 +369,7 @@ $q = \chern^{\beta_0}_1(u) = c - r\beta_0$,
we get:
\begin{sagesilent}
from plots_and_expressions import bgmlv2_with_q
from plots_and_expressions import bgmlv2_with_q
\end{sagesilent}
\begin{equation}
\sage{bgmlv2_with_q}
......@@ -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}.
\begin{sagesilent}
from plots_and_expressions import bgmlv2_d_ineq
from plots_and_expressions import bgmlv2_d_ineq
\end{sagesilent}
\begin{equation}
\label{eqn-bgmlv2_d_upperbound}
......@@ -400,7 +401,7 @@ $d$ yields:
\begin{sagesilent}
from plots_and_expressions import bgmlv3_d_upperbound_terms
from plots_and_expressions import bgmlv3_d_upperbound_terms
\end{sagesilent}
\begin{equation}
......@@ -433,31 +434,31 @@ see-saw principle.
% TODO maybe cover the see-saw principle
\begin{align*}
\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
&=
& =
\left(
\mu(v-u) - \beta_0
\mu(v-u) - \beta_0
\right)^2
\\
&>
\\
& >
\left(
\mu(v) - \beta_0
\mu(v) - \beta_0
\right)^2
&\text{by Equation \ref{lem:proof:slope-order-rltR}}
\\
&=
& \text{by Equation \ref{lem:proof:slope-order-rltR}}
\\
& =
\left(
\frac{\chern^{\beta_0}_1(v)}{\chern_0(v)}
\frac{\chern^{\beta_0}_1(v)}{\chern_0(v)}
\right)^2
\\
&\geq
\\
& \geq
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{So}
\quad
\\
\text{So}
\quad
\frac{
\left(
q-\chern^{\beta_0}_1(v)
......@@ -467,9 +468,9 @@ see-saw principle.
R-r
\right)^2
}
&>
& >
2 \frac{\chern^{\beta_0}_2(v)}{R}
&
&
\text{and}
\quad
\chern_2^{\beta_0}(v)
......@@ -482,7 +483,7 @@ see-saw principle.
R-r
\right)
}
&<
& <
\frac{r\chern^{\beta_0}_2(v)}{R}
\end{align*}
\noindent
......@@ -493,7 +494,7 @@ are greater than those of Equation
\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}
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
\ref{eqn:u-coords}, amounts to the following:
\begin{sagesilent}
from plots_and_expressions import bgmlv2_d_upperbound_terms
from plots_and_expressions import bgmlv2_d_upperbound_terms
\end{sagesilent}
\begin{align}
d &>
d & >
\frac{1}{2}{\beta_0}^2 r
+ {\beta_0} q,
\phantom{+} % to keep terms aligned
&\qquad\text{when\:} r > 0
& \qquad\text{when\:} r > 0
\label{eqn:radiuscond_d_bound_betamin}
\\
d &\leq
\\
d & \leq
\sage{bgmlv2_d_upperbound_terms.problem2.linear}
+ \sage{bgmlv2_d_upperbound_terms.problem2.const}
+\sage{bgmlv2_d_upperbound_terms.problem2.hyperbolic},
&\qquad\text{when\:} r > 0
\label{eqn:bgmlv2_d_bound_betamin}
\\
d &\leq
+\sage{bgmlv2_d_upperbound_terms.problem2.hyperbolic},
& \qquad\text{when\:} r > 0
\label{eqn:bgmlv2_d_bound_betamin}
\\
d & \leq
\sage{bgmlv3_d_upperbound_terms.problem2.linear}
+ \sage{bgmlv3_d_upperbound_terms.problem2.const}
% ^ ch_2^\beta(F)=0 for beta_{-}
\sage{bgmlv3_d_upperbound_terms.problem2.hyperbolic},
&\qquad\text{when\:} r > R
\label{eqn:bgmlv3_d_bound_betamin}
\sage{bgmlv3_d_upperbound_terms.problem2.hyperbolic},
& \qquad\text{when\:} r > R
\label{eqn:bgmlv3_d_bound_betamin}
\end{align}
Recalling that $q \coloneqq \chern^{\beta}_1(u) \in (0, \chern^{\beta}_1(v))$,
......@@ -560,22 +561,22 @@ This will be pursued in Subsection
\ref{subsec:bounds-on-semistab-rank-prob-2}.
\begin{sagesilent}
from plots_and_expressions import typical_bounds_on_d
from plots_and_expressions import typical_bounds_on_d
\end{sagesilent}
\begin{figure}
\centering
\sageplot[width=\linewidth]{typical_bounds_on_d}
\caption{
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 $q\coloneqq\chern_1^{\beta}(u)$.
Where $\chern(v) = (3,2\ell,-2\ell^2)$.
}
\label{fig:d_bounds_xmpl_gnrc_q}
\centering
\sageplot[width=\linewidth]{typical_bounds_on_d}
\caption{
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 $q\coloneqq\chern_1^{\beta}(u)$.
Where $\chern(v) = (3,2\ell,-2\ell^2)$.
}
\label{fig:d_bounds_xmpl_gnrc_q}
\end{figure}
\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}
Unlike for Problem \ref{problem:problem-statement-2},
......@@ -588,37 +589,37 @@ bounds do not share the same assymptote as the lower bound
\begin{align}
\sage{problem1.radius_condition_d_bound.lhs()}
&>
& >
\sage{problem1.radius_condition_d_bound.rhs()}
&\text{when }r>0
& \text{when }r>0
\label{eqn:prob1:radiuscond}
\\
d &\leq
d & \leq
\sage{problem1.bgmlv2_d_upperbound_terms.linear}
+ \sage{problem1.bgmlv2_d_upperbound_terms.const}
+ \sage{problem1.bgmlv2_d_upperbound_terms.hyperbolic}
&\text{when }r>R
& \text{when }r>R
\label{eqn:prob1:bgmlv2}
\\
d &\leq
d & \leq
\sage{problem1.bgmlv3_d_upperbound_terms.linear}
+ \sage{problem1.bgmlv3_d_upperbound_terms.const}
\sage{problem1.bgmlv3_d_upperbound_terms.hyperbolic}
&\text{when }r>R
\sage{problem1.bgmlv3_d_upperbound_terms.hyperbolic}
& \text{when }r>R
\label{eqn:prob1:bgmlv3}
\end{align}
\begin{figure}
\centering
\sageplot[width=\linewidth]{problem1.example_plot}
\caption{
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 $q\coloneqq\chern_1^{\beta}(E)$.
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
\ref{problem:problem-statement-1}.
}
\label{fig:problem1:d_bounds_xmpl_gnrc_q}
\centering
\sageplot[width=\linewidth]{problem1.example_plot}
\caption{
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 $q\coloneqq\chern_1^{\beta}(E)$.
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
\ref{problem:problem-statement-1}.
}
\label{fig:problem1:d_bounds_xmpl_gnrc_q}
\end{figure}
......@@ -630,11 +631,11 @@ This is because $R\coloneqq\chern_0(v)$
and $\chern_2^{\beta_0}(v)$ are all strictly positive:
\begin{itemize}
\item $R > 0$ from the setting of Problem
\ref{problem:problem-statement-1}
\ref{problem:problem-statement-1}
\item $\chern_2^{\beta_0}(v)>0$
by Lemma \ref{lem:comparison-test-with-beta_}
because ${\beta_0} < \beta_{-}$ due to the choice of $P$ being
a point on $\Theta_v^{-}$
by Lemma \ref{lem:comparison-test-with-beta_}
because ${\beta_0} < \beta_{-}$ due to the choice of $P$ being
a point on $\Theta_v^{-}$
\end{itemize}
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
idea will be pursued in Subsection \ref{subsec:bounds-on-semistab-rank-prob-1}.
\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}
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 $d$
\ref{fig:problem1:d_bounds_xmpl_gnrc_q}).
\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}
and $q\coloneqq\chern_1^{B}(u)$.
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 $d$
Solving for the lower bound in Equation \ref{eqn:prob1:radiuscond} being
less than the upper bound in Equation \ref{eqn:prob1:bgmlv2} yields:
\begin{equation}
r<\sage{problem1.positive_intersection_bgmlv2}
r<\sage{problem1.positive_intersection_bgmlv2}
\end{equation}
\noindent
Similarly, but with the upper bound in Equation \ref{eqn:prob1:bgmlv3}, gives:
\begin{equation}
r<\sage{problem1.positive_intersection_bgmlv3}
r<\sage{problem1.positive_intersection_bgmlv3}
\end{equation}
\noindent
......@@ -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}.
\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}.
Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression:
\begin{equation}
......@@ -731,7 +732,7 @@ following Lemma \ref{lem:prob1:convenient_r_bound}.
\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}
Now, the inequalities from the above Subsubsection
......@@ -763,13 +764,13 @@ $\chern^{\beta_0}(u) > 0$
(Equation \ref{eqn:radiuscond_d_bound_betamin}) we get:
\begin{sagesilent}
from plots_and_expressions import \
positive_radius_condition_with_q, \
q_value_expr, \
beta_value_expr
from plots_and_expressions import \
positive_radius_condition_with_q, \
q_value_expr, \
beta_value_expr
\end{sagesilent}
\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
\ni
\:\:
......@@ -786,10 +787,10 @@ proof of Theorem
\begin{sagesilent}
from plots_and_expressions import main_theorem1, betamin_subs
from plots_and_expressions import main_theorem1, betamin_subs
\end{sagesilent}
\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
line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
$m\coloneqq\ell^2$.
......@@ -803,8 +804,8 @@ from plots_and_expressions import main_theorem1, betamin_subs
\begin{align*}
\min
\left(
\sage{main_theorem1.r_upper_bound1.subs(betamin_subs)}, \:\:
\sage{main_theorem1.r_upper_bound2.subs(betamin_subs)}
\sage{main_theorem1.r_upper_bound1.subs(betamin_subs)}, \:\:
\sage{main_theorem1.r_upper_bound2.subs(betamin_subs)}
\right)
\end{align*}
\noindent
......@@ -812,57 +813,57 @@ from plots_and_expressions import main_theorem1, betamin_subs
\end{theorem}
\begin{proof}
Both $d$ and the lower bound in
(Equation \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
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,
then there are no solutions for $d$.
Hence any corresponding $r$ cannot be a rank of a
pseudo-semistabiliser for $v$.
To avoid this, we must have,
considering Equations
\ref{eqn:bgmlv2_d_bound_betamin},
\ref{eqn:bgmlv3_d_bound_betamin},
\ref{eqn:radiuscond_d_bound_betamin}.
Both $d$ and the lower bound in
(Equation \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
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,
then there are no solutions for $d$.
Hence any corresponding $r$ cannot be a rank of a
pseudo-semistabiliser for $v$.
To avoid this, we must have,
considering Equations
\ref{eqn:bgmlv2_d_bound_betamin},
\ref{eqn:bgmlv3_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}
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}
\noindent
This is equivalent to:
\noindent
This is equivalent to:
\begin{equation}
\label{eqn:thm-bound-for-r-impossible-cond-for-r}
r \leq
\min\left(
\begin{equation}
\label{eqn:thm-bound-for-r-impossible-cond-for-r}
r \leq
\min\left(
\sage{
main_theorem1.r_upper_bound1
} ,
\sage{
main_theorem1.r_upper_bound2
}
\right)
\end{equation}
\right)
\end{equation}
\end{proof}
\begin{sagesilent}
from plots_and_expressions import q_sol, bgmlv_v, psi
from plots_and_expressions import q_sol, bgmlv_v, psi
\end{sagesilent}
\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
line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
$m\coloneqq\ell^2$.
......@@ -873,90 +874,90 @@ from plots_and_expressions import q_sol, bgmlv_v, psi
are bounded above as follows.
\begin{align*}
r &\leq \sage{main_theorem1.corollary_r_bound}
&\text{if } R < \frac{\Delta(v)\lcm(m,2n^2)}{2m}
r & \leq \sage{main_theorem1.corollary_r_bound}
& \text{if } R < \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}
r & \leq \frac{\Delta(v)\lcm(m,2n^2)}{2m}
& \text{if } R \geq \frac{\Delta(v)\lcm(m,2n^2)}{2m}
\end{align*}
\end{corollary}
\begin{proof}
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
\ref{thm:rmax_with_uniform_eps}.
Noticing that the expression is a maximum of two quadratic functions in $q$
($\beta_0=\beta_{-}(v)$ in this context):
\begin{equation*}
f_1(q)\coloneqq\sage{main_theorem1.r_upper_bound1} \qquad
f_2(q)\coloneqq\sage{main_theorem1.r_upper_bound2}
\end{equation*}
These have their minimums at $q=0$ and $q=\chern_1^{\beta_{-}}(v)$ respectively,
with values 0 and $R>0$ respectively.
So provided that
$f_2\left(\chern^{\beta_{-}}_1(v)\right) < f_1\left(\chern^{\beta_{-}}_1(v)\right)$,
the maximum is achieved at their intersection.
Otherwise, the maximum is achieved at
$\chern^{\beta_{-}}_1(v)$.
So we can say that
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
\ref{thm:rmax_with_uniform_eps}.
Noticing that the expression is a maximum of two quadratic functions in $q$
($\beta_0=\beta_{-}(v)$ in this context):
\begin{equation*}
f_1(q)\coloneqq\sage{main_theorem1.r_upper_bound1} \qquad
f_2(q)\coloneqq\sage{main_theorem1.r_upper_bound2}
\end{equation*}
These have their minimums at $q=0$ and $q=\chern_1^{\beta_{-}}(v)$ respectively,
with values 0 and $R>0$ respectively.
So provided that
$f_2\left(\chern^{\beta_{-}}_1(v)\right) < f_1\left(\chern^{\beta_{-}}_1(v)\right)$,
the maximum is achieved at their intersection.
Otherwise, the maximum is achieved at
$\chern^{\beta_{-}}_1(v)$.
So we can say that
\begin{align*}
r &\leq
\begin{align*}
r & \leq
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)
\\ &&
\\ &&
\text{where $q_{\mathrm{max}}$ is the $q$-value where the $f_i$ intersect}
\\
r &\leq f_1\left(\chern^{\beta_{-}}(v)\right)
&\text{if } f_2\left(\chern^{\beta_{-}}_1(v)\right) \geq
\\
r & \leq f_1\left(\chern^{\beta_{-}}(v)\right)
& \text{if } f_2\left(\chern^{\beta_{-}}_1(v)\right) \geq
f_1\left(\chern^{\beta_{-}}_1(v)\right)
\end{align*}
\end{align*}
\noindent
In the first case,
solving for $f_1(q)=f_2(q)$ yields
\begin{equation*}
q=\sage{q_sol.expand()}
\end{equation*}
And evaluating $f_1$ at this $q$-value gives:
\begin{equation*}
\sage{main_theorem1.corollary_intermediate}
\end{equation*}
\noindent
In the first case,
solving for $f_1(q)=f_2(q)$ yields
\begin{equation*}
q=\sage{q_sol.expand()}
\end{equation*}
And evaluating $f_1$ at this $q$-value gives:
\begin{equation*}
\sage{main_theorem1.corollary_intermediate}
\end{equation*}
\noindent
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.
\noindent
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.
\end{proof}
\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-second}
Just like in Example \ref{exmpl:recurring-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{recurring.betaminus}$,
giving $n=\sage{recurring.n}$.
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
$\sage{recurring.corrolary_bound} \approx
\sage{round(float(recurring.corrolary_bound), 1)}$,
which is much closer to real maximum 25 than the original bound 144.
\label{exmpl:recurring-second}
Just like in Example \ref{exmpl:recurring-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{recurring.betaminus}$,
giving $n=\sage{recurring.n}$.
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
$\sage{recurring.corrolary_bound} \approx
\sage{round(float(recurring.corrolary_bound), 1)}$,
which is much closer to real maximum 25 than the original bound 144.
\end{example}
\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-second}
Just like in Example \ref{exmpl:extravagant-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{extravagant.betaminus}$,
giving $n=\sage{extravagant.n}$.
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
$\sage{extravagant.corrolary_bound} \approx
\sage{round(float(extravagant.corrolary_bound), 1)}$,
which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
original bound 215296.
\label{exmpl:extravagant-second}
Just like in Example \ref{exmpl:extravagant-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{extravagant.betaminus}$,
giving $n=\sage{extravagant.n}$.
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
$\sage{extravagant.corrolary_bound} \approx
\sage{round(float(extravagant.corrolary_bound), 1)}$,
which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
original bound 215296.
\end{example}
%% 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
proof of Theorem \ref{thm:rmax_with_uniform_eps}:
\begin{lemmadfn}[%
A better alternative to $\epsilon_v$:
$\epsilon_{v,q}$
A better alternative to $\epsilon_v$:
$\epsilon_{v,q}$
]
\label{lemdfn:epsilon_q}
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}:
\mod{\gcd\left(
\frac{n^2\gcd(m,2)}{\gcd(m,2n^2)},
\frac{mn\aa}{\gcd(m,2n^2)}
\right)}
\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.
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(
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(
\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 $k_{v,q}=1$.
\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
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}
\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
$m\coloneqq\ell^2$.
......@@ -1122,8 +1123,8 @@ from plots_and_expressions import main_theorem2
\begin{align*}
\min
\left(
\sage{main_theorem2.r_upper_bound1.subs(betamin_subs)}, \:\:
\sage{main_theorem2.r_upper_bound2.subs(betamin_subs)}
\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},
......@@ -1142,10 +1143,10 @@ 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
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}
\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 $\beta_{-}\coloneqq\beta_{-}(v)=\frac{a_v}{n}$
......@@ -1158,127 +1159,127 @@ from plots_and_expressions import main_theorem2_corollary
\begin{align*}
\min
\left(
\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_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
${
k \equiv -\aa\bb
\pmod{n}
}$.
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 $n$.
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}
\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.
\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}
\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}.
\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}
tex/computing-solutions.tex
View file @ 1a27cd54
......@@ -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.
\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
semistabilisers, as given by Theorem \ref{thm:loose-bound-on-r}.
......@@ -36,7 +36,7 @@ the Bogomolov inequalities and Consequence 3 of Lemma
($\chern_2^{\beta_{-}}(u)>0$).
\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
$\chern_2^{\beta_{-}}(u)>0$ condition induces a lower bound on $d$.
......@@ -51,8 +51,8 @@ end up not yielding any solutions for the problem.
In fact, solutions $u$ to our problem with high rank must have $\mu(u)$ close to
$\beta_{-}(v)$:
\begin{align*}
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 \chern_1^{\beta_{-}}(u) \leq \chern_1^{\beta_{-}}(u) \\
0 & \leq \mu(u) - \beta_{-} \leq \frac{\chern_1^{\beta_{-}}(v)}{r}
\end{align*}
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 $\beta_{-}$.
......@@ -66,17 +66,17 @@ alternative algorithm which will later be described in Section
\ref{sect:prob2-algorithm}.
\begin{center}
\label{table:bench-schmidt-vs-nay}
\begin{tabular}{ |r|l|l| }
\hline
Choice of $v$ on $\mathbb{P}^2$
& $(3, 2\ell, -2)$
& $(3, 2\ell, -\frac{15}{2})$ \\
\hline
\cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s & >1hr \\
\cite{NaylorRust2023} exec time & \sim 50ms & \sim 50ms \\
\hline
\end{tabular}
\label{table:bench-schmidt-vs-nay}
\begin{tabular}{ |r|l|l| }
\hline
Choice of $v$ on $\mathbb{P}^2$
& $(3, 2\ell, -2)$
& $(3, 2\ell, -\frac{15}{2})$ \\
\hline
\cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s & >1hr \\
\cite{NaylorRust2023} exec time & \sim 50ms & \sim 50ms \\
\hline
\end{tabular}
\end{center}
\section{Computing Solutions to Problem \ref{problem:problem-statement-2}}
......@@ -94,7 +94,7 @@ The algorithm yields solutions
$u=(r,c\ell,d\ell^2)$ to the problem as follows.
\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
$v$ has positive rank, that it satisfies $\Delta(v) \geq 0$, and that
......@@ -102,6 +102,7 @@ $\beta_{-}(v)$ is rational.
Take $\beta_{-}(v)=\frac{a_v}{n}$ in simplest terms.
Iterate over $q = \frac{b_q}{n} \in (0,\chern_1^{\beta_{-}}(v))\cap\frac{1}{n}\ZZ$.
The code used to generate the corresponding values for $b_q$ is shown in Listing
% texlab: ignore
\ref{fig:code:consideredb}.
\lstinputlisting[
......@@ -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 $q=\chern_1^\beta(u)$
(i.e. choice of $b$).
The code representing this appears in Listing
% texlab: ignore
\ref{fig:code:reducingtoeachb}.
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
......@@ -133,9 +135,9 @@ and collect up the results.
]{../tilt.rs/src/tilt_stability/find_all.git-untrack.rs.tex.git-untrack}
\subsubsection{Iterating Over Possible
\texorpdfstring{$r=\chern_0(u)$}{r}
for Fixed
\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
\texorpdfstring{$r=\chern_0(u)$}{r}
for Fixed
\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
}
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
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
for $q$ and $r$ (and consequently $c$) considered.
% texlab: ignore
This is shown in Listing \ref{fig:code:reducingtoeachr}.
\lstinputlisting[
......@@ -170,11 +173,11 @@ This is shown in Listing \ref{fig:code:reducingtoeachr}.
\subsubsection{Iterating Over Possible
\texorpdfstring{$d=\chern_2(u)/\ell^2$}{d}
for Fixed
\texorpdfstring{$r=\chern_0(u)$}{r}
and
\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
\texorpdfstring{$d=\chern_2(u)/\ell^2$}{d}
for Fixed
\texorpdfstring{$r=\chern_0(u)$}{r}
and
\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
}
At this point we are considering a specialisation of the problem
......@@ -198,8 +201,10 @@ equivalent to bounds on $d$ given by the equations
in Subsubsection \ref{subsubsect:all-bounds-on-d-prob2}
It therefore remains to just pick values
$d\in\frac{1}{\lcm(m,2n^2)}\ZZ$ within the bounds.
% texlab: ignore
Listing \ref{fig:code:solveforfixedr} is the code for solving this
specialisation of the problem, where the possible $d$ values are computed in
% texlab: ignore
Listing \ref{fig:code:possible_chern2}.
The explicit code for the bounds can be found in Appendix
\ref{appendix:subsubsec:fixed-r}.
......@@ -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:
\begin{itemize}
\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 Modern CPU architecture such as branch predictors
\cite{BranchPredictor2024} may offset the overhead of considering ranks that
turn out to be too large to have any solutions.
\cite{BranchPredictor2024} may offset the overhead of considering ranks that
turn out to be too large to have any solutions.
\end{itemize}
For relatively small Chern characters (as those appearing in examples so far),
......