diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index a1ff9fdc3faf21e4efaf2f71c122c66c351ca0f1..a45a5f17302cc21d95bc58b6bc280d3422f83da7 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -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}
diff --git a/tex/computing-solutions.tex b/tex/computing-solutions.tex
index 96763db414a03b2bc13ae7edc56b6a44e53c21a7..690dbdf98e7ed407cec76f0069c282eb3ffff77d 100644
--- a/tex/computing-solutions.tex
+++ b/tex/computing-solutions.tex
@@ -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),