From 15b090c01c268d450e33d30315c8085dc6210337 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 2 Jul 2024 16:58:07 +0100
Subject: [PATCH] Add diagram for Schmidt argument
---
figures/schmidt-arg-diag.pdf | Bin 0 -> 4116 bytes
figures/schmidt-arg-diag.pdf_tex | 61 +++++++
figures/schmidt-arg-diag.svg | 262 ++++++++++++++++++++++++++++++
tex/bounds-on-semistabilisers.tex | 50 +++---
4 files changed, 353 insertions(+), 20 deletions(-)
create mode 100644 figures/schmidt-arg-diag.pdf
create mode 100644 figures/schmidt-arg-diag.pdf_tex
create mode 100644 figures/schmidt-arg-diag.svg
diff --git a/figures/schmidt-arg-diag.pdf b/figures/schmidt-arg-diag.pdf
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diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index 372e135..39dfa81 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -37,30 +37,40 @@ bounded above by:
\begin{proof}
The Bogomolov form applied to the twisted Chern character is the same as the
-untwisted one. So $0 \leq \Delta(u)$
-(condition 2 from Corollary \ref{cor:num_test_prob2})
-yields:
-
-\begin{equation}
- \label{eqn-bgmlv-on-E}
- 2\chern^\beta_0(u) \chern^\beta_2(u) \leq \chern^\beta_1(u)^2
-\end{equation}
+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}
-% FUTURE maybe ref this back to some definition of first tilt
+\begin{minipage}{0.59\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^\beta_0(u) \chern^\beta_2(u) \leq \chern^\beta_1(u)^2
+ \end{equation}
-\noindent
-The restrictions on $\chern^{\beta_-}_0(u)$ and $\chern^{\beta_-}_2(v)$
-is best seen with the following graph:
+ \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}
-% TODO: hyperbola restriction graph (shaded)
+ \noindent
+ The restrictions on $\chern^{\beta_-}_0(u)$ and $\chern^{\beta_-}_2(v)$
+ is best seen with the following graph:
+ % TODO: hyperbola restriction graph (shaded)
+\end{minipage}
+\hfill
+\begin{minipage}{0.39\linewidth}
+ %\label{prop:proof:fig:pseudowall-pos}
+ \begin{center}
+ \def\svgwidth{\linewidth}
+ \subimport{../figures/}{schmidt-arg-diag.pdf_tex}
+ \end{center}
+ \caption{Restrictions}
+\end{minipage}
This is where the rationality of $\beta_{-}$ comes in. If
--
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