diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index 30b8bfe73738c9def660fa4a0a18e4ed4420e4e2..d8d7366c03629cca32b27a749238c58e25da5d55 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -1,6 +1,4 @@
-\section{B.Schmidt's Solutions to the Problems}
-
-\subsection{Bound on \texorpdfstring{$\chern_0(u)$}{ch0(u)} for Semistabilizers}
+\section{Existing Bound on Semistabiliser Ranks}
\label{subsect:loose-bound-on-r}
The proof for the following Theorem \ref{thm:loose-bound-on-r} was hinted at in
@@ -8,23 +6,28 @@ The proof for the following Theorem \ref{thm:loose-bound-on-r} was hinted at in
\cite{SchmidtGithub2020}. The latter reference is a SageMath \cite{sagemath}
library for computing certain quantities related to Bridgeland stabilities on
Picard rank 1 varieties. It also includes functions to compute pseudo-walls and
-pseudo-semistabilizers for tilt stability.
+pseudo-semistabilisers for tilt stability.
\begin{theorem}[Bound on $r$ - Benjamin Schmidt]
\label{thm:loose-bound-on-r}
-Given a Chern character $v$ such that $\beta_-\coloneqq\beta_{-}(v)\in\QQ$, the rank $r$ of
-any semistabilizer $E$ of some $F \in \firsttilt{\beta_-}$ with $\chern(F)=v$ is
+Given a Chern character $v$ with $\Delta(v) \geq 0$ and positive rank (or
+$\chern_0(v) = 0$ but $\chern_1(v) > 0$)
+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{mn^2 \chern^{\beta_-}_1(v)^2}{\gcd(m,2n^2)}
+ r \leq \frac{m\left(n \chern^{\beta_-}_1(v) - 1\right)^2}{\gcd(m,2n^2)}
\end{equation*}
\end{theorem}
\begin{proof}
The Bogomolov form applied to the twisted Chern character is the same as the
-normal one. So $0 \leq \Delta(E)$ yields:
+untwisted one. So $0 \leq \Delta(E)$
+(condition 2 from Corollary \ref{cor:num_test_prob2})
+yields:
\begin{equation}
\label{eqn-bgmlv-on-E}
@@ -32,7 +35,9 @@ normal one. So $0 \leq \Delta(E)$ yields:
\end{equation}
\noindent
-Furthermore, $E \hookrightarrow F$ in $\firsttilt{\beta_{-}}$ gives:
+Furthermore,
+condition 5 from Corollary \ref{cor:num_test_prob2}
+gives:
\begin{equation}
\label{eqn-tilt-cat-cond}
0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)
@@ -58,8 +63,27 @@ bound for the rank of $E$:
\chern_0(E) &= \chern^{\beta_-}_0(E) \\
&\leq \frac{\lcm(m,2n^2) \chern^{\beta_-}_1(E)^2}{2} \\
&= \frac{mn^2 \chern^{\beta_-}_1(F)^2}{\gcd(m,2n^2)}
+ \label{proof:first-bound-on-r}
\end{align}
+In fact Equation \ref{eqn-tilt-cat-cond} can be tightened slightly:
+we cannot have equality $\chern^{\beta_{-}}_1(E) = \chern^{\beta_{-}}_1(F)$
+otherwise we would have $\chern^{\beta_{-}}_1(G)=0$ for the quotient $G$.
+This would imply $\mu(G)=\beta_{-}$, but since $\Theta_G$ is bounded above in the
+upper-half plane by the assymptotes crossing the $\beta$-axis at 45$^\circ$ at
+$\beta=\beta_{-}(v)$. So $\Theta_G$ cannot intersect $\Theta_v$ at any point
+with $\alpha > 0$, so there is no point with $\nu(E)=\nu(F)=\nu(G)=0$, which would
+have to hold at the top of the pseudo-wall if it were to exist.
+Therefore we must have a strict inequality
+$\chern^{\beta_{-}}_1(E) < \chern^{\beta_{-}}_1(F)$,
+and since these are elements of $\frac{1}{n}\ZZ$, we can also conclude:
+\[
+ n\chern^{\beta_{-}}_1(E) \leq n\chern^{\beta_{-}}_1(F) - 1
+\]
+which then tightens the upper bound found for $\chern_0(E)$
+in Equation \ref{proof:first-bound-on-r}
+to the bound in the statement of the Lemma.
+
\end{proof}
\begin{sagesilent}
@@ -74,8 +98,8 @@ 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 semistabilizers for $v$ are bounded above by $\sage{recurring.loose_bound}$.
-However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
+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}
@@ -92,8 +116,8 @@ 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 semistabilizers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
-However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
+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 $\sage{extravagant.actual_rmax}$.
\end{example}
@@ -101,7 +125,7 @@ rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
\section{Tighter Bounds}
\label{sec:refinement}
-To get tighter bounds on the rank of destabilizers $E$ of some $F$ with some
+To get tighter bounds on the rank of destabilisers $E$ of some $F$ with some
fixed Chern character, we will need to consider each of the values which
$\chern_1^{\beta}(E)$ can take.
Doing this will allow us to eliminate possible values of $\chern_0(E)$ for which
@@ -111,7 +135,7 @@ corresponding $\chern_1^{\beta}(E)$ fail one of the inequalities (which is what
was implicitly happening before).
-First, let us fix a Chern character for $F$, and some pseudo-semistabilizer
+First, let us fix a Chern character for $F$, and some pseudo-semistabiliser
$u$ which is a solution to problem
\ref{problem:problem-statement-1} or
\ref{problem:problem-statement-2}.
@@ -154,7 +178,7 @@ For the next subsections, we consider $q$ to be fixed with one of these values,
and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
-\subsection{Numerical Inequalities}
+\subsection{Bounds on \texorpdfstring{`$d$'}{d}-values for Solutions of Problems}
This section studies the numerical conditions that $u$ must satisfy as per
lemma \ref{lem:num_test_prob1}
@@ -388,7 +412,7 @@ it is worth noting that the extreme values of $q$ in this range lead to the
tightest bounds on $d$, as illustrated in Figure
(\ref{fig:d_bounds_xmpl_extrm_q}).
In fact, in each case, one of the weak upper bounds coincides with one of the
-weak lower bounds, (implying no possible destabilizers $E$ with
+weak lower bounds, (implying no possible destabilisers $E$ with
$\chern_0(E)=\vcentcolon r>R\coloneqq\chern_0(F)$ for these $q$-values).
This indeed happens in general since the right hand sides of
(eqn \ref{eqn:bgmlv2_d_bound_betamin}) and
@@ -401,7 +425,7 @@ In the other case, $q=\chern^{\beta}_1(F)$, it is the right hand sides of
The more generic case, when $0 < q\coloneqq\chern_1^{\beta}(E) < \chern_1^{\beta}(F)$
for the bounds on $d$ in terms of $r$ is illustrated in Figure
(\ref{fig:d_bounds_xmpl_gnrc_q}).
-The question of whether there are pseudo-destabilizers of arbitrarily large
+The question of whether there are pseudo-destabilisers of arbitrarily large
rank, in the context of the graph, comes down to whether there are points
$(r,d) \in \ZZ \oplus \frac{1}{\lcm(m,2)} \ZZ$ (with large $r$)
% TODO have a proper definition for pseudo-destabilizers/walls
@@ -631,7 +655,7 @@ from plots_and_expressions import main_theorem1
\begin{theorem}[Bound on $r$ \#1]
\label{thm:rmax_with_uniform_eps}
Let $v = (R,C\ell,D\ell^2)$ be a fixed Chern character. Then the ranks of the
- pseudo-semistabilizers for $v$,
+ pseudo-semistabilisers for $v$,
which are solutions to problem \ref{problem:problem-statement-2},
with $\chern_1^\beta = q$
are bounded above by the following expression.
@@ -660,7 +684,7 @@ So, if any of the two upper bounds on $d$ come to within
$\frac{1}{\lcm(m,2n^2)}$ of this lower bound, then there are no solutions for
$d$.
Hence any corresponding $r$ cannot be a rank of a
-pseudo-semistabilizer for $v$.
+pseudo-semistabiliser for $v$.
To avoid this, we must have,
considering Equations
@@ -706,7 +730,7 @@ from plots_and_expressions import q_sol, bgmlv_v, psi
\label{cor:direct_rmax_with_uniform_eps}
Let $v$ be a fixed Chern character and
$R\coloneqq\chern_0(v) \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
- Then the ranks of the pseudo-semistabilizers for $v$,
+ Then the ranks of the pseudo-semistabilisers for $v$,
which are solutions to problem \ref{problem:problem-statement-2},
are bounded above by the following expression.
@@ -716,7 +740,7 @@ from plots_and_expressions import q_sol, bgmlv_v, psi
\end{corollary}
\begin{proof}
-The ranks of the pseudo-semistabilizers for $v$ are bounded above by the
+The ranks of the pseudo-semistabilisers for $v$ are bounded above by the
maximum over $q\in [0, \chern_1^{\beta}(F)]$ of the expression in theorem
\ref{thm:rmax_with_uniform_eps}.
Noticing that the expression is a maximum of two quadratic functions in $q$:
@@ -753,7 +777,7 @@ 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 semistabilizers for $v$ are bounded above by
+the ranks of tilt semistabilisers for $v$ are bounded above by
$\sage{recurring.corrolary_bound} \approx \sage{float(recurring.corrolary_bound)}$,
which is much closer to real maximum 25 than the original bound 144.
\end{example}
@@ -766,7 +790,7 @@ 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 semistabilizers for $v$ are bounded above by
+the ranks of tilt semistabilisers for $v$ are bounded above by
$\sage{extravagant.corrolary_bound} \approx \sage{float(extravagant.corrolary_bound)}$,
which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
original bound 215296.
@@ -784,9 +808,9 @@ this value of $\frac{1}{2n^2}\ZZ$ explicitly.
The expressions that will follow will be a bit more complicated and have more
parts which depend on the values of $q$ and $\beta$, even their numerators
$\aa,\bb$ specifically. The upcoming Theorem (TODO ref) is less useful as a
-`clean' formula for a bound on the ranks of the pseudo-semistabilizers, but has a
+`clean' formula for a bound on the ranks of the pseudo-semistabilisers, but has a
purpose in the context of writing a computer program to find
-pseudo-semistabilizers. Such a program would iterate through possible values of
+pseudo-semistabilisers. Such a program would iterate through possible values of
$q$, then iterate through values of $r$ within the bounds (dependent on $q$),
which would then determine $c$, and then find the corresponding possible values
for $d$.
@@ -928,7 +952,7 @@ from plots_and_expressions import main_theorem2
\label{thm:rmax_with_eps1}
Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
rational and expressed in lowest terms.
- Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with,
+ Then the ranks $r$ of the pseudo-semistabilisers $u$ for $v$ with,
which are solutions to problem \ref{problem:problem-statement-2},
$\chern_1^\beta(u) = q = \frac{b_q}{n}$
are bounded above by the following expression:
@@ -959,7 +983,7 @@ from plots_and_expressions import main_theorem2_corollary
(or any other surfaces with $m=1$ or $2$).
Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
rational and expressed in lowest terms.
- Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with,
+ Then the ranks $r$ of the pseudo-semistabilisers $u$ for $v$ with,
which are solutions to problem \ref{problem:problem-statement-2},
$\chern_1^\beta(u) = q = \frac{b_q}{n}$
are bounded above by the following expression:
@@ -997,7 +1021,7 @@ 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 semistabilizer $u$ of $v$
+The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabiliser $u$ of $v$
in terms of the possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
\begin{sagesilent}
@@ -1032,7 +1056,7 @@ end}
\noindent
It's 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-semistabilizers of $v$ in this case.
+pseudo-semistabilisers of $v$ in this case.
As a reminder, the original loose bound from Theorem \ref{thm:loose-bound-on-r}
was 144.
@@ -1050,7 +1074,7 @@ possible values for $k_{v,q}$, in dfn/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 semistabilizer $u$ of $v$
+The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabiliser $u$ of $v$
in terms of the first few smallest possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
\begin{sagesilent}