From 16c6732abde238900b4f4146a402b6ceb0b7b7a2 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 18 Jul 2023 15:27:47 +0100
Subject: [PATCH] Tweak := to be more centered
---
main.tex | 51 ++++++++++++++++++++++++++-------------------------
1 file changed, 26 insertions(+), 25 deletions(-)
diff --git a/main.tex b/main.tex
index 73f2fc5..b4bf7a9 100644
--- a/main.tex
+++ b/main.tex
@@ -11,6 +11,7 @@
\usepackage{minted}
\usepackage{subcaption}
\usepackage{cancel}
+\usepackage{mathtools}
\usepackage[]{breqn}
\usepackage[
backend=biber,
@@ -538,7 +539,7 @@ normal one. So $0 \leq \Delta(E)$ yields:
\begin{theorem}[Bound on $r$ - Benjamin Schmidt]
\label{thm:loose-bound-on-r}
-Given a Chern character $v$ such that $\beta_-:=\beta_{-}(v)\in\QQ$, the rank $r$ of
+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
bounded above by:
@@ -748,8 +749,8 @@ was implicitly happening before).
First, let us fix a Chern character for $F$, and some semistabilizer $E$:
\begin{align}
- v &:= \chern(F) = (R,C\ell,D\ell^2) \\
- u &:= \chern(E) = (r,c\ell,d\ell^2)
+ v &\coloneqq \chern(F) = (R,C\ell,D\ell^2) \\
+ u &\coloneqq \chern(E) = (r,c\ell,d\ell^2)
\end{align}
\begin{sagesilent}
@@ -783,7 +784,7 @@ c_in_terms_of_q = c_lower_bound + q
\begin{equation}
\label{eqn-cintermsofm}
c=\chern_1(u) = \sage{c_in_terms_of_q}
- \qquad 0 \leq q := \chern_1^{\beta}(u) \leq \chern_1^{\beta}(v)
+ \qquad 0 \leq q \coloneqq \chern_1^{\beta}(u) \leq \chern_1^{\beta}(v)
\end{equation}
Furthermore, $\chern_1 \in \ZZ$ so we only need to consider
@@ -851,7 +852,7 @@ in the context of our problem.
Finally, $r>0$ as per the statement of the problem, so the right-hand-side
of equation \ref{eqn:bgmlv1-pt1} is always greater than, or equal, to zero.
-And so, when $P:=(\beta_{-},0)$, this condition $\Delta(u,v-u) \geq 0$ is
+And so, when $P\coloneqq(\beta_{-},0)$, this condition $\Delta(u,v-u) \geq 0$ is
always satisfied when $2r \geq R$, provided that the other conditions of the
problem statement (\ref{subsect:problem-statement}) hold.
@@ -1334,20 +1335,20 @@ def plot_d_bound(
\label{fig:d_bounds_xmpl_max_q}
\end{subfigure}
\caption{
- Bounds on $d:=\chern_2(E)$ in terms of $r:=\chern_0(E)$ for fixed, extreme,
- values of $q:=\chern_1^{\beta}(E)$.
+ Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq\chern_0(E)$ for fixed, extreme,
+ values of $q\coloneqq\chern_1^{\beta}(E)$.
Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_extrm_q}
\end{figure}
-Recalling that $q := \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
+Recalling that $q \coloneqq \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
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
-$\chern_0(E)=:r>R:=\chern_0(F)$ for these $q$-values).
+$\chern_0(E)=\vcentcolon r>R\coloneqq\chern_0(F)$ for these $q$-values).
This indeed happens in general since the right hand sides of
(eqn \ref{eqn:bgmlv2_d_bound_betamin}) and
(eqn \ref{eqn:positive_rad_d_bound_betamin}) match when $q=0$.
@@ -1356,7 +1357,7 @@ In the other case, $q=\chern^{\beta}_1(F)$, it is the right hand sides of
(eqn \ref{eqn:positive_rad_d_bound_betamin}) which match.
-The more generic case, when $0 < q:=\chern_1^{\beta}(E) < \chern_1^{\beta}(F)$
+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
@@ -1379,8 +1380,8 @@ Some of the details around the associated numerics are explored next.
width=\linewidth
]{plot_d_bound(v_example, 2, ymax=4, ymin=-2, aspect_ratio=1)}
\caption{
- Bounds on $d:=\chern_2(E)$ in terms of $r:=\chern_0(E)$ for a fixed
- value $\chern_1^{\beta}(F)/2$ of $q:=\chern_1^{\beta}(E)$.
+ Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq \chern_0(E)$ for a fixed
+ value $\chern_1^{\beta}(F)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_gnrc_q}
@@ -1399,7 +1400,7 @@ $(r,c,d)$ that satisfy all inequalities to give a pseudowall.
The strategy here is similar to what was shown in (sect
\ref{sec:twisted-chern}).
-One specialization here is to use that $\ell:=c_1(H)$ generates $NS(X)$, so that
+One specialization here is to use that $\ell\coloneqq c_1(H)$ generates $NS(X)$, so that
in fact, any Chern character can be written as
$\left(r,c\ell,\frac{e}{2}\ell^2\right)$ for $r,c,e\in\ZZ$.
% ref to Schmidt?
@@ -1415,7 +1416,7 @@ q_value_expr = (q == b_q/n)
Suppose $\beta = \frac{\aa}{n}$ for some coprime $n \in \NN,\aa \in \ZZ$.
Then fix a value of $q$:
\begin{equation}
- q:=\chern_1^{\beta}(E)
+ q\coloneqq \chern_1^{\beta}(E)
=\frac{\bb}{n}
\in
\frac{1}{n} \ZZ
@@ -1530,7 +1531,7 @@ bounds_too_tight_condition2 = (
\sage{bgmlv2_d_upperbound_exp_term},
\sage{bgmlv3_d_upperbound_exp_term_alt2}
\right)
- \geq \epsilon := \frac{1}{2n^2}
+ \geq \epsilon \coloneqq \frac{1}{2n^2}
\end{equation}
\egroup
@@ -1576,7 +1577,7 @@ r_upper_bound_all_q = (
\begin{corrolary}[Bound on $r$ \#2]
\label{cor:direct_rmax_with_uniform_eps}
Let $v$ be a fixed Chern character and
- $R:=\chern_0(v) \leq n^2\Delta(v)$.
+ $R\coloneqq\chern_0(v) \leq n^2\Delta(v)$.
Then the ranks of the pseudo-semistabilizers for $v$
are bounded above by the following expression.
@@ -1599,8 +1600,8 @@ 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 $q$:
\begin{equation*}
- f_1(q):=\sage{r_upper_bound1.subs(kappa==1).rhs()} \qquad
- f_2(q):=\sage{r_upper_bound2.subs(kappa==1).rhs()}
+ f_1(q)\coloneqq\sage{r_upper_bound1.subs(kappa==1).rhs()} \qquad
+ f_2(q)\coloneqq\sage{r_upper_bound2.subs(kappa==1).rhs()}
\end{equation*}
These have their minimums at $q=0$ and $q=\chern_1^{\beta}(F)$ respectively.
It suffices to find their intersection in
@@ -1687,7 +1688,7 @@ which would then determine $c$, and then find the corresponding possible values
for $d$.
-Firstly, we only need to consider $r$-values for which $c:=\chern_1(E)$ is
+Firstly, we only need to consider $r$-values for which $c\coloneqq\chern_1(E)$ is
integral:
\begin{equation}
@@ -1742,7 +1743,7 @@ proof of theorem \ref{thm:rmax_with_uniform_eps}:
Where $\epsilon_{v,q}$ is defined as follows:
\begin{equation*}
- \epsilon_{v,q} :=
+ \epsilon_{v,q} \coloneqq
\frac{k_{q}}{2n^2}
\end{equation*}
with $k_{v,q}$ being the least $k\in\ZZ_{>0}$ satisfying $k \equiv -\aa\bb \mod n$
@@ -1800,7 +1801,7 @@ $\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.
\begin{theorem}[Bound on $r$ \#3]
\label{thm:rmax_with_eps1}
- Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta:=\beta(v)$
+ 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
$\chern_1^\beta(u) = q = \frac{b_q}{n}$
@@ -1837,8 +1838,8 @@ take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
$\beta=\sage{recurring.b}$, giving $n=\sage{recurring.b.denominator()}$
and $\chern_1^{\sage{recurring.b}}(F) = \sage{recurring.twisted.ch[1]}$.
%% TODO transcode notebook code
-The (non-exclusive) upper bounds for $r:=\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
-in terms of the possible values for $q:=\chern_1^{\beta}(u)$ are as follows:
+The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
+in terms of the possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
\begin{sagesilent}
import numpy as np
@@ -1923,8 +1924,8 @@ 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 $\sage{n}$ smaller, for any given $q$ value.
-The (non-exclusive) upper bounds for $r:=\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
-in terms of the first few smallest possible values for $q:=\chern_1^{\beta}(u)$ are as follows:
+The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ 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)
--
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